Theorem antisymrelressn 39118

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 38785	. 2 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
2		relres 5972	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfantisymrel5 39116	. 2 ⊢ ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	mpbir2an 712	1 ⊢ AntisymRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  {csn 4582   class class class wbr 5100   ↾ cres 5634  Rel wrel 5637   AntisymRel wantisymrel 38473

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-cnvrefrel 38858  df-antisymrel 39114

This theorem is referenced by: (None)