Theorem brcosscnvcoss 39233

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 1894	. . 3 ⊢ (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
3		brcoss 39230	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4		brcoss 39230	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
5	4	ancoms 464	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)))
6	2, 3, 5	3bitr4d 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2146   class class class wbr 5111   ≀ ccoss 38892

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-coss 39210

This theorem is used by:  cocossss  39235  cnvcosseq  39236  rncossdmcoss  39254  symrelcoss3  39264  eleccossin  39282