Theorem eleccossin 38506

Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐶 have the common element 𝐵. (Contributed by Peter Mazsa, 15-Oct-2021.)

Assertion

Ref	Expression
eleccossin	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶)))

Proof of Theorem eleccossin

Step	Hyp	Ref	Expression
1		elin 3947	. . 3 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅))
2		relcoss 38446	. . . . 5 ⊢ Rel ≀ 𝑅
3		relelec 8771	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵))
4	2, 3	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵)
5		relelec 8771	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵))
6	2, 5	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵)
7	4, 6	anbi12i 628	. . 3 ⊢ ((𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
8	1, 7	bitri 275	. 2 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
9		brcosscnvcoss 38457	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐶 ↔ 𝐶 ≀ 𝑅𝐵))
10	9	anbi2d 630	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵)))
11	8, 10	bitr4id 290	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∩ cin 3930   class class class wbr 5124  Rel wrel 5664  [cec 8722   ≀ ccoss 38204

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8726  df-coss 38434

This theorem is referenced by:  trcoss2  38507