Theorem antisymrelressn 39371

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.

Step	Hyp	Ref	Expression
1		antisymressn 39038	. 2 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
2		relres 5993	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfantisymrel5 39369	. 2 ⊢ ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	mpbir2an 721	1 ⊢ AntisymRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562  {csn 4584   class class class wbr 5102   ↾ cres 5651  Rel wrel 5654   AntisymRel wantisymrel 38726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-cnvrefrel 39111  df-antisymrel 39367

This theorem is referenced by: (None)