Theorem symrelcoss3 37848

Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 37933. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)

Assertion

Ref	Expression
symrelcoss3	⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)

Proof of Theorem symrelcoss3

Step	Hyp	Ref	Expression
1		brcosscnvcoss 37817	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥))
2	1	el2v 3476	. . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)
3	2	biimpi 215	. . 3 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
4	3	gen2 1790	. 2 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)
5		relcoss 37806	. 2 ⊢ Rel ≀ 𝑅
6	4, 5	pm3.2i 470	1 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  Vcvv 3468   class class class wbr 5141  Rel wrel 5674   ≀ ccoss 37556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-coss 37794

This theorem is referenced by:  symrelcoss2  37849  eqvrelcoss3  38001