Theorem splitelof-P6-L1 777

Description: Lemma for splitelof-P6 778.

Assertion

Ref	Expression
splitelof-P6-L1	⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎   𝑡,𝑏   𝑢,𝑏,𝑎

Proof of Theorem splitelof-P6-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. . . . 5 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑡 ∈ 𝑢) → (𝑎 = 𝑡 ∧ 𝑏 = 𝑢))
2		subelofd-P5.CL 643	. . . . . . 7 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑎 ∈ 𝑏 ↔ 𝑡 ∈ 𝑢))
3	2	ndbier-P3.15 180	. . . . . 6 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑡 ∈ 𝑢 → 𝑎 ∈ 𝑏))
4	3	import-P3.34a.RC 306	. . . . 5 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑡 ∈ 𝑢) → 𝑎 ∈ 𝑏)
5	1, 4	ndandi-P3.7 172	. . . 4 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑡 ∈ 𝑢) → ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
6		exi-P6 718	. . . 4 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → ∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
7		exi-P6 718	. . . 4 ⊢ (∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
8	5, 6, 7	dsyl-P3.25.RC 262	. . 3 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑡 ∈ 𝑢) → ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏))
9	8	rcp-NDIMI2 224	. 2 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑡 ∈ 𝑢 → ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))
10	2	ndbief-P3.14 179	. . . . . 6 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑎 ∈ 𝑏 → 𝑡 ∈ 𝑢))
11	10	import-P3.34a.RC 306	. . . . 5 ⊢ (((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → 𝑡 ∈ 𝑢)
12	11	exiav-P5 615	. . . 4 ⊢ (∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → 𝑡 ∈ 𝑢)
13	12	exiav-P5 615	. . 3 ⊢ (∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → 𝑡 ∈ 𝑢)
14	13	rcp-NDIMP0addall 207	. 2 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏) → 𝑡 ∈ 𝑢))
15	9, 14	ndbii-P3.13 178	1 ⊢ ((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) → (𝑡 ∈ 𝑢 ↔ ∃𝑎∃𝑏((𝑎 = 𝑡 ∧ 𝑏 = 𝑢) ∧ 𝑎 ∈ 𝑏)))

Syntax hints:  term-obj 1   = wff-equals 6   ∈ wff-elemof 7   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L8-inl 20  ax-L8-inr 21  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  splitelof-P6  778