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Theorem disjne 4455
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 4450 . . 3 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 eleq1 2829 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
32notbid 318 . . . . 5 (𝑥 = 𝐶 → (¬ 𝑥𝐵 ↔ ¬ 𝐶𝐵))
43rspccva 3621 . . . 4 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → ¬ 𝐶𝐵)
5 eleq1a 2836 . . . . 5 (𝐷𝐵 → (𝐶 = 𝐷𝐶𝐵))
65necon3bd 2954 . . . 4 (𝐷𝐵 → (¬ 𝐶𝐵𝐶𝐷))
74, 6syl5com 31 . . 3 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → (𝐷𝐵𝐶𝐷))
81, 7sylanb 581 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → (𝐷𝐵𝐶𝐷))
983impia 1118 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-dif 3954  df-in 3958  df-nul 4334
This theorem is referenced by:  brdom7disj  10571  brdom6disj  10572  frlmssuvc1  21814  f1resrcmplf1dlem  35100  kelac1  43075
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