Theorem disjne 3597

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 3592	. . 3 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B)
2		eleq1 2413	. . . . . 6 ⊢ (x = C → (x ∈ B ↔ C ∈ B))
3	2	notbid 285	. . . . 5 ⊢ (x = C → (¬ x ∈ B ↔ ¬ C ∈ B))
4	3	rspccva 2955	. . . 4 ⊢ ((∀x ∈ A ¬ x ∈ B ∧ C ∈ A) → ¬ C ∈ B)
5		eleq1a 2422	. . . . 5 ⊢ (D ∈ B → (C = D → C ∈ B))
6	5	necon3bd 2554	. . . 4 ⊢ (D ∈ B → (¬ C ∈ B → C ≠ D))
7	4, 6	syl5com 26	. . 3 ⊢ ((∀x ∈ A ¬ x ∈ B ∧ C ∈ A) → (D ∈ B → C ≠ D))
8	1, 7	sylanb 458	. 2 ⊢ (((A ∩ B) = ∅ ∧ C ∈ A) → (D ∈ B → C ≠ D))
9	8	3impia 1148	1 ⊢ (((A ∩ B) = ∅ ∧ C ∈ A ∧ D ∈ B) → C ≠ D)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)