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Theorem disjne 4376
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 4371 . . 3 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 eleq1 2901 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
32notbid 321 . . . . 5 (𝑥 = 𝐶 → (¬ 𝑥𝐵 ↔ ¬ 𝐶𝐵))
43rspccva 3597 . . . 4 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → ¬ 𝐶𝐵)
5 eleq1a 2909 . . . . 5 (𝐷𝐵 → (𝐶 = 𝐷𝐶𝐵))
65necon3bd 3025 . . . 4 (𝐷𝐵 → (¬ 𝐶𝐵𝐶𝐷))
74, 6syl5com 31 . . 3 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → (𝐷𝐵𝐶𝐷))
81, 7sylanb 584 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → (𝐷𝐵𝐶𝐷))
983impia 1114 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  cin 3907  c0 4265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-ral 3135  df-dif 3911  df-in 3915  df-nul 4266
This theorem is referenced by:  brdom7disj  9942  brdom6disj  9943  frlmssuvc1  20481  frlmsslsp  20483  f1resrcmplf1dlem  32433  kelac1  39937
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