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

Theorem compsscnvlem 9075
Description: Lemma for compsscnv 9076. (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 3699 . . . 4 (𝐴𝑥) ⊆ 𝐴
31, 2syl6eqss 3618 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦𝐴)
4 selpw 4115 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
53, 4sylibr 223 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 ∈ 𝒫 𝐴)
61difeq2d 3690 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴𝑦) = (𝐴 ∖ (𝐴𝑥)))
7 elpwi 4117 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
87adantr 480 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥𝐴)
9 dfss4 3820 . . . 4 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
108, 9sylib 207 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴 ∖ (𝐴𝑥)) = 𝑥)
116, 10eqtr2d 2645 . 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 1475  wcel 1977  cdif 3537  wss 3540  𝒫 cpw 4108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110
This theorem is referenced by:  compsscnv  9076
  Copyright terms: Public domain W3C validator