Theorem disjne 4406

Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
disjne	⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Proof of Theorem disjne

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 4401	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		eleq1 2849	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
3	2	notbid 320	. . . . 5 ⊢ (𝑥 = 𝐶 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵))
4	3	rspccva 3579	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
5		eleq1a 2856	. . . . 5 ⊢ (𝐷 ∈ 𝐵 → (𝐶 = 𝐷 → 𝐶 ∈ 𝐵))
6	5	necon3bd 2970	. . . 4 ⊢ (𝐷 ∈ 𝐵 → (¬ 𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐷))
7	4, 6	syl5com 31	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷))
8	1, 7	sylanb 590	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷))
9	8	3impia 1129	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ∩ cin 3901  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-dif 3905  df-in 3909  df-nul 4284

This theorem is referenced by:  brdom7disj  10481  brdom6disj  10482  frlmssuvc1  21833  f1resrcmplf1dlem  35340  kelac1  43600