Theorem eleccossin 38746

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 3917	. . 3 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅))
2		relcoss 38686	. . . . 5 ⊢ Rel ≀ 𝑅
3		relelec 8682	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵))
4	2, 3	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵)
5		relelec 8682	. . . . 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 38697	. . 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 2113   ∩ cin 3900   class class class wbr 5098  Rel wrel 5629  [cec 8633   ≀ ccoss 38383

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-coss 38674

This theorem is referenced by:  trcoss2  38747