Theorem compsscnvlem 10455

Description: Lemma for compsscnv 10456. (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 490	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 = (𝐴 ∖ 𝑥))
2		difss 4071	. . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
3	1, 2	eqsstrdi 3953	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ⊆ 𝐴)
4		velpw 4561	. . 3 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
5	3, 4	sylibr 237	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ∈ 𝒫 𝐴)
6	1	difeq2d 4062	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝐴 ∖ 𝑥)))
7		elpwi 4563	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
8	7	adantr 486	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 ⊆ 𝐴)
9		dfss4 4213	. . . 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 2796	. 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 = (𝐴 ∖ 𝑦))
12	5, 11	jca 521	1 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-pw 4558

This theorem is used by:  compsscnv  10456