Theorem eleccossin 39072

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 3920	. . 3 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅))
2		relcoss 39012	. . . . 5 ⊢ Rel ≀ 𝑅
3		relelec 8726	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵))
4	2, 3	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵)
5		relelec 8726	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵))
6	2, 5	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐶] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅𝐵)
7	4, 6	anbi12i 637	. . 3 ⊢ ((𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
8	1, 7	bitri 277	. 2 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵))
9		brcosscnvcoss 39023	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐶 ↔ 𝐶 ≀ 𝑅𝐵))
10	9	anbi2d 639	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐶 ≀ 𝑅𝐵)))
11	8, 10	bitr4id 292	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142   ∩ cin 3903   class class class wbr 5100  Rel wrel 5652  [cec 8676   ≀ ccoss 38682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-coss 39000

This theorem is referenced by:  trcoss2  39073