Theorem eleccossin 36601

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 3903	. . 3 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅))
2		relcoss 36546	. . . . 5 ⊢ Rel ≀ 𝑅
3		relelec 8543	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵))
4	2, 3	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵)
5		relelec 8543	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵))
6	2, 5	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵)
7	4, 6	anbi12i 627	. . 3 ⊢ ((𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
8	1, 7	bitri 274	. 2 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
9		brcosscnvcoss 36557	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐶 ↔ 𝐶 ≀ 𝑅𝐵))
10	9	anbi2d 629	. 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 205   ∧ wa 396   ∈ wcel 2106   ∩ cin 3886   class class class wbr 5074  Rel wrel 5594  [cec 8496   ≀ ccoss 36333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-coss 36537

This theorem is referenced by:  trcoss2  36602