Theorem ecin0 38334

Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.)

Assertion

Ref	Expression
ecin0	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem ecin0

Step	Hyp	Ref	Expression
1		disj1 4458	. 2 ⊢ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2		elecg 8788	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
3	2	el2v1 38204	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
4	3	adantr 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
5		elecALTV 38248	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
6	5	elvd 3484	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
7	6	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
8	7	notbid 318	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
9	4, 8	imbi12d 344	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
10	9	albidv 1918	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
11	1, 10	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962  ∅c0 4339   class class class wbr 5148  [cec 8742

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-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-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-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746

This theorem is referenced by:  ecinn0  38335