Theorem eleccossin 38465

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 3979	. . 3 ⊢ (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅 ∧ 𝐵 ∈ [𝐶] ≀ 𝑅))
2		relcoss 38405	. . . . 5 ⊢ Rel ≀ 𝑅
3		relelec 8791	. . . . 5 ⊢ (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵))
4	2, 3	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐵)
5		relelec 8791	. . . . 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 38416	. . 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 2106   ∩ cin 3962   class class class wbr 5148  Rel wrel 5694  [cec 8742   ≀ ccoss 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-coss 38393

This theorem is referenced by:  trcoss2  38466