Theorem antisymrelressn 39719

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 39386	. 2 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
2		relres 5992	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfantisymrel5 39717	. 2 ⊢ ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	mpbir2an 724	1 ⊢ AntisymRel (𝑅 ↾ {𝐴})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  {csn 4583   class class class wbr 5102   ↾ cres 5649  Rel wrel 5652   AntisymRel wantisymrel 39074

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-cnvrefrel 39459  df-antisymrel 39715

This theorem is used by: (None)