Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  antisymressn Structured version   Visualization version   GIF version

Theorem antisymressn 37968
Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37966) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
Assertion
Ref Expression
antisymressn 𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)

Proof of Theorem antisymressn
StepHypRef Expression
1 brressn 37965 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦)))
21el2v 3471 . . . 4 (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦))
32simplbi 496 . . 3 (𝑥(𝑅 ↾ {𝐴})𝑦𝑥 = 𝐴)
4 brressn 37965 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥)))
54el2v 3471 . . . 4 (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥))
65simplbi 496 . . 3 (𝑦(𝑅 ↾ {𝐴})𝑥𝑦 = 𝐴)
7 eqtr3 2751 . . 3 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
83, 6, 7syl2an 594 . 2 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
98gen2 1790 1 𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  Vcvv 3463  {csn 4625   class class class wbr 5144  cres 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-xp 5679  df-res 5685
This theorem is referenced by:  antisymrelressn  38288
  Copyright terms: Public domain W3C validator