Theorem ecin0 39284

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 4405	. 2 ⊢ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2		elecg 8762	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
3	2	el2v1 39161	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
4	3	adantr 486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
5		elecALTV 39203	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
6	5	elvd 3457	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
7	6	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
8	7	notbid 321	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
9	4, 8	imbi12d 347	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
10	9	albidv 1953	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
11	1, 10	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  ∅c0 4279   class class class wbr 5103  [cec 8715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719

This theorem is used by:  ecinn0  39285