Theorem antisymressn 38430

Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38428) 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 38427	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
2	1	el2v 3457	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
3	2	simplbi 497	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 → 𝑥 = 𝐴)
4		brressn 38427	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥)))
5	4	el2v 3457	. . . 4 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥))
6	5	simplbi 497	. . 3 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 → 𝑦 = 𝐴)
7		eqtr3 2752	. . 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 1796	1 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  Vcvv 3450  {csn 4591   class class class wbr 5109   ↾ cres 5642

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-xp 5646  df-res 5652

This theorem is referenced by:  antisymrelressn  38751