Theorem brcosscnvcoss 38425

Description: For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.)

Assertion

Ref	Expression
brcosscnvcoss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

Proof of Theorem brcosscnvcoss

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exancom 1861	. . 3 ⊢ (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
3		brcoss 38422	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4		brcoss 38422	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
5	4	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
6	2, 3, 5	3bitr4d 311	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   class class class wbr 5107   ≀ ccoss 38169

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-coss 38402

This theorem is referenced by:  cocossss  38427  cnvcosseq  38428  rncossdmcoss  38446  symrelcoss3  38456  eleccossin  38474