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Theorem dmsnn0 5335
 Description: The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmsnn0

Proof of Theorem dmsnn0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2959 . . . . 5
21eldm 5067 . . . 4
3 df-br 4213 . . . . . 6
4 opex 4427 . . . . . . 7
54elsnc 3837 . . . . . 6
6 eqcom 2438 . . . . . 6
73, 5, 63bitri 263 . . . . 5
87exbii 1592 . . . 4
92, 8bitr2i 242 . . 3
109exbii 1592 . 2
11 elvv 4936 . 2
12 n0 3637 . 2
1310, 11, 123bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wb 177  wex 1550   wceq 1652   wcel 1725   wne 2599  cvv 2956  c0 3628  csn 3814  cop 3817   class class class wbr 4212   cxp 4876   cdm 4878 This theorem is referenced by:  rnsnn0  5336  dmsn0  5337  dmsn0el  5339  relsn2  5340  1st2val  6372  mpt2xopxnop0  6466  hashfun  11700  1stnpr  24093 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888
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