Theorem ecin0 38719

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 4380	. 2 ⊢ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2		elecg 8678	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
3	2	el2v1 38596	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
4	3	adantr 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
5		elecALTV 38638	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
6	5	elvd 3437	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
7	6	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
8	7	notbid 319	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
9	4, 8	imbi12d 345	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
10	9	albidv 1927	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
11	1, 10	bitrid 284	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882  ∅c0 4261   class class class wbr 5072  [cec 8631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635

This theorem is referenced by:  ecinn0  38720