Theorem antisymrelressn 38751

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 38430	. 2 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
2		relres 5978	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfantisymrel5 38749	. 2 ⊢ ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	mpbir2an 711	1 ⊢ AntisymRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  {csn 4591   class class class wbr 5109   ↾ cres 5642  Rel wrel 5645   AntisymRel wantisymrel 38201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-cnvrefrel 38513  df-antisymrel 38747

This theorem is referenced by: (None)