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Theorem compsscnvlem 9230
Description: Lemma for compsscnv 9231. (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
compsscnvlem ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem compsscnvlem
StepHypRef Expression
1 simpr 476 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 = (𝐴𝑥))
2 difss 3770 . . . 4 (𝐴𝑥) ⊆ 𝐴
31, 2syl6eqss 3688 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦𝐴)
4 selpw 4198 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
53, 4sylibr 224 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 ∈ 𝒫 𝐴)
61difeq2d 3761 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴𝑦) = (𝐴 ∖ (𝐴𝑥)))
7 elpwi 4201 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
87adantr 480 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥𝐴)
9 dfss4 3891 . . . 4 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
108, 9sylib 208 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴 ∖ (𝐴𝑥)) = 𝑥)
116, 10eqtr2d 2686 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥 = (𝐴𝑦))
125, 11jca 553 1 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cdif 3604  wss 3607  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by:  compsscnv  9231
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