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

Theorem idssen 8550
 Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
idssen I ⊆ ≈

Proof of Theorem idssen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5685 . 2 Rel I
2 vex 3483 . . . . 5 𝑦 ∈ V
32ideq 5710 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
4 eqeng 8539 . . . . 5 (𝑥 ∈ V → (𝑥 = 𝑦𝑥𝑦))
54elv 3485 . . . 4 (𝑥 = 𝑦𝑥𝑦)
63, 5sylbi 220 . . 3 (𝑥 I 𝑦𝑥𝑦)
7 df-br 5053 . . 3 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8 df-br 5053 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
96, 7, 83imtr3i 294 . 2 (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
101, 9relssi 5647 1 I ⊆ ≈
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  Vcvv 3480   ⊆ wss 3919  ⟨cop 4556   class class class wbr 5052   I cid 5446   ≈ cen 8502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-en 8506 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator