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Theorem disjne 3596
 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) = C A D B) → CD)

Proof of Theorem disjne
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 disj 3591 . . 3 ((AB) = x A ¬ x B)
2 eleq1 2413 . . . . . 6 (x = C → (x BC B))
32notbid 285 . . . . 5 (x = C → (¬ x B ↔ ¬ C B))
43rspccva 2954 . . . 4 ((x A ¬ x B C A) → ¬ C B)
5 eleq1a 2422 . . . . 5 (D B → (C = DC B))
65necon3bd 2553 . . . 4 (D B → (¬ C BCD))
74, 6syl5com 26 . . 3 ((x A ¬ x B C A) → (D BCD))
81, 7sylanb 458 . 2 (((AB) = C A) → (D BCD))
983impia 1148 1 (((AB) = C A D B) → CD)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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