MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  compsscnvlem Structured version   Visualization version   GIF version

Theorem compsscnvlem 9795
Description: Lemma for compsscnv 9796. (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 487 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 = (𝐴𝑥))
2 difss 4111 . . . 4 (𝐴𝑥) ⊆ 𝐴
31, 2eqsstrdi 4024 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦𝐴)
4 velpw 4547 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
53, 4sylibr 236 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 ∈ 𝒫 𝐴)
61difeq2d 4102 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴𝑦) = (𝐴 ∖ (𝐴𝑥)))
7 elpwi 4551 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
87adantr 483 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥𝐴)
9 dfss4 4238 . . . 4 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
108, 9sylib 220 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴 ∖ (𝐴𝑥)) = 𝑥)
116, 10eqtr2d 2860 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥 = (𝐴𝑦))
125, 11jca 514 1 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  cdif 3936  wss 3939  𝒫 cpw 4542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-in 3946  df-ss 3955  df-pw 4544
This theorem is referenced by:  compsscnv  9796
  Copyright terms: Public domain W3C validator