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Theorem antisymrelressn 37157
Description: (Contributed by Peter Mazsa, 29-Jun-2024.)
Assertion
Ref Expression
antisymrelressn AntisymRel (𝑅 ↾ {𝐴})

Proof of Theorem antisymrelressn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 antisymressn 36837 . 2 𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
2 relres 5964 . 2 Rel (𝑅 ↾ {𝐴})
3 dfantisymrel5 37155 . 2 ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥𝑦((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴})))
41, 2, 3mpbir2an 709 1 AntisymRel (𝑅 ↾ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  {csn 4584   class class class wbr 5103  cres 5633  Rel wrel 5636   AntisymRel wantisymrel 36602
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-cnvrefrel 36920  df-antisymrel 37153
This theorem is referenced by: (None)
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