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Theorem antisymressn 37314
Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37312) 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 37311 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦)))
21el2v 3483 . . . 4 (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦))
32simplbi 499 . . 3 (𝑥(𝑅 ↾ {𝐴})𝑦𝑥 = 𝐴)
4 brressn 37311 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥)))
54el2v 3483 . . . 4 (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥))
65simplbi 499 . . 3 (𝑦(𝑅 ↾ {𝐴})𝑥𝑦 = 𝐴)
7 eqtr3 2759 . . 3 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
83, 6, 7syl2an 597 . 2 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
98gen2 1799 1 𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  Vcvv 3475  {csn 4629   class class class wbr 5149  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689
This theorem is referenced by:  antisymrelressn  37634
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