Theorem compsscnvlem 10283

Description: Lemma for compsscnv 10284. (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 485	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 = (𝐴 ∖ 𝑥))
2		difss 4066	. . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
3	1, 2	eqsstrdi 3959	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ⊆ 𝐴)
4		velpw 4534	. . 3 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
5	3, 4	sylibr 235	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ∈ 𝒫 𝐴)
6	1	difeq2d 4057	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝐴 ∖ 𝑥)))
7		elpwi 4536	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
8	7	adantr 481	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 ⊆ 𝐴)
9		dfss4 4197	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
10	8, 9	sylib 219	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
11	6, 10	eqtr2d 2775	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 = (𝐴 ∖ 𝑦))
12	5, 11	jca 516	1 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-pw 4531

This theorem is referenced by:  compsscnv  10284