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Theorem disjne 3411
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 3406 . . 3 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 eleq1 2200 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
32notbid 656 . . . . 5 (𝑥 = 𝐶 → (¬ 𝑥𝐵 ↔ ¬ 𝐶𝐵))
43rspccva 2783 . . . 4 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → ¬ 𝐶𝐵)
5 eleq1a 2209 . . . . 5 (𝐷𝐵 → (𝐶 = 𝐷𝐶𝐵))
65necon3bd 2349 . . . 4 (𝐷𝐵 → (¬ 𝐶𝐵𝐶𝐷))
74, 6syl5com 29 . . 3 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → (𝐷𝐵𝐶𝐷))
81, 7sylanb 282 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → (𝐷𝐵𝐶𝐷))
983impia 1178 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  wne 2306  wral 2414  cin 3065  c0 3358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359
This theorem is referenced by: (None)
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