Theorem antisymressn 38556

Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38554) 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 38553	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
2	1	el2v 3443	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
3	2	simplbi 497	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 → 𝑥 = 𝐴)
4		brressn 38553	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥)))
5	4	el2v 3443	. . . 4 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥))
6	5	simplbi 497	. . 3 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 → 𝑦 = 𝐴)
7		eqtr3 2753	. . 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 1797	1 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  Vcvv 3436  {csn 4573   class class class wbr 5089   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-res 5626

This theorem is referenced by:  antisymrelressn  38872