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Theorem disjne 4402
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 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)

Proof of Theorem disjne
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 disj 4397 . . 3 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 eleq1 2900 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
32notbid 319 . . . . 5 (𝑥 = 𝐶 → (¬ 𝑥𝐵 ↔ ¬ 𝐶𝐵))
43rspccva 3621 . . . 4 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → ¬ 𝐶𝐵)
5 eleq1a 2908 . . . . 5 (𝐷𝐵 → (𝐶 = 𝐷𝐶𝐵))
65necon3bd 3030 . . . 4 (𝐷𝐵 → (¬ 𝐶𝐵𝐶𝐷))
74, 6syl5com 31 . . 3 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → (𝐷𝐵𝐶𝐷))
81, 7sylanb 581 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → (𝐷𝐵𝐶𝐷))
983impia 1109 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3016  wral 3138  cin 3934  c0 4290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-dif 3938  df-in 3942  df-nul 4291
This theorem is referenced by:  brdom7disj  9942  brdom6disj  9943  frlmssuvc1  20868  frlmsslsp  20870  f1resrcmplf1dlem  32257  kelac1  39543
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