Theorem antisymressn 38440

Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38438) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)

Assertion

Ref	Expression
antisymressn	⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)

Proof of Theorem antisymressn

Step	Hyp	Ref	Expression
1		brressn 38437	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
2	1	el2v 3488	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
3	2	simplbi 497	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 → 𝑥 = 𝐴)
4		brressn 38437	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥)))
5	4	el2v 3488	. . . 4 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥))
6	5	simplbi 497	. . 3 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 → 𝑦 = 𝐴)
7		eqtr3 2763	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
8	3, 6, 7	syl2an 596	. 2 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
9	8	gen2 1795	1 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537   = wceq 1539  Vcvv 3481  {csn 4634   class class class wbr 5151   ↾ cres 5695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-xp 5699  df-res 5705

This theorem is referenced by:  antisymrelressn  38760