Theorem compsscnvlem 10349

Description: Lemma for compsscnv 10350. (Contributed by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
compsscnvlem	⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem compsscnvlem

Step	Hyp	Ref	Expression
1		simpr 489	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 = (𝐴 ∖ 𝑥))
2		difss 4090	. . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
3	1, 2	eqsstrdi 3981	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ⊆ 𝐴)
4		velpw 4567	. . 3 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
5	3, 4	sylibr 237	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ∈ 𝒫 𝐴)
6	1	difeq2d 4081	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝐴 ∖ 𝑥)))
7		elpwi 4569	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
8	7	adantr 485	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 ⊆ 𝐴)
9		dfss4 4222	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
10	8, 9	sylib 221	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
11	6, 10	eqtr2d 2799	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 = (𝐴 ∖ 𝑦))
12	5, 11	jca 520	1 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-pw 4564

This theorem is referenced by:  compsscnv  10350