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Theorem dmsnn0 5138
 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 2791 . . . . 5
21eldm 4876 . . . 4
3 df-br 4024 . . . . . 6
4 opex 4237 . . . . . . 7
54elsnc 3663 . . . . . 6
6 eqcom 2285 . . . . . 6
73, 5, 63bitri 262 . . . . 5
87exbii 1569 . . . 4
92, 8bitr2i 241 . . 3
109exbii 1569 . 2
11 elvv 4748 . 2
12 n0 3464 . 2
1310, 11, 123bitr4i 268 1
 Colors of variables: wff set class Syntax hints:   wb 176  wex 1528   wceq 1623   wcel 1684   wne 2446  cvv 2788  c0 3455  csn 3640  cop 3643   class class class wbr 4023   cxp 4687   cdm 4689 This theorem is referenced by:  rnsnn0  5139  dmsn0  5140  dmsn0el  5142  relsn2  5143  1st2val  6145  hashfun  11389  1stnpr  23245  imfstnrelc  25081  mpt2xopxnop0  28081 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699
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