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Theorem disjne 3858
 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

Proof of Theorem disjne
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 disj 3853 . . 3
2 eleq1 2515 . . . . . 6
32notbid 294 . . . . 5
43rspccva 3195 . . . 4
5 eleq1a 2526 . . . . 5
65necon3bd 2655 . . . 4
74, 6syl5com 30 . . 3
81, 7sylanb 472 . 2
983impia 1194 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   w3a 974   wceq 1383   wcel 1804   wne 2638  wral 2793   cin 3460  c0 3770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-v 3097  df-dif 3464  df-in 3468  df-nul 3771 This theorem is referenced by:  brdom7disj  8912  brdom6disj  8913  frlmssuvc1  18802  frlmssuvc1OLD  18804  frlmsslsp  18806  frlmsslspOLD  18807  kelac1  30984
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