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Theorem antisymressn 37840
Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37838) 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 37837 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦)))
21el2v 3477 . . . 4 (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦))
32simplbi 497 . . 3 (𝑥(𝑅 ↾ {𝐴})𝑦𝑥 = 𝐴)
4 brressn 37837 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥)))
54el2v 3477 . . . 4 (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴𝑦𝑅𝑥))
65simplbi 497 . . 3 (𝑦(𝑅 ↾ {𝐴})𝑥𝑦 = 𝐴)
7 eqtr3 2753 . . 3 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
83, 6, 7syl2an 595 . 2 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
98gen2 1791 1 𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532   = wceq 1534  Vcvv 3469  {csn 4624   class class class wbr 5142  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-res 5684
This theorem is referenced by:  antisymrelressn  38160
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