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Theorem disjne 3273
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 3268 . . 3 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 eleq1 2100 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
32notbid 592 . . . . 5 (𝑥 = 𝐶 → (¬ 𝑥𝐵 ↔ ¬ 𝐶𝐵))
43rspccva 2655 . . . 4 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → ¬ 𝐶𝐵)
5 eleq1a 2109 . . . . 5 (𝐷𝐵 → (𝐶 = 𝐷𝐶𝐵))
65necon3bd 2248 . . . 4 (𝐷𝐵 → (¬ 𝐶𝐵𝐶𝐷))
74, 6syl5com 26 . . 3 ((∀𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴) → (𝐷𝐵𝐶𝐷))
81, 7sylanb 268 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → (𝐷𝐵𝐶𝐷))
983impia 1101 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  wne 2204  wral 2306  cin 2916  c0 3224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225
This theorem is referenced by: (None)
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