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Theorem dmsnn0 5478
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 3112 . . . . 5
21eldm 5205 . . . 4
3 df-br 4453 . . . . . 6
4 opex 4716 . . . . . . 7
54elsnc 4053 . . . . . 6
6 eqcom 2466 . . . . . 6
73, 5, 63bitri 271 . . . . 5
87exbii 1667 . . . 4
92, 8bitr2i 250 . . 3
109exbii 1667 . 2
11 elvv 5063 . 2
12 n0 3794 . 2
1310, 11, 123bitr4i 277 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652   cvv 3109   c0 3784  {csn 4029  <.cop 4035   class class class wbr 4452  X.cxp 5002  domcdm 5004
This theorem is referenced by:  rnsnn0  5479  dmsn0  5480  dmsn0el  5482  relsn2  5483  1stnpr  6804  1st2val  6826  mpt2xopxnop0  6962  hashfun  12495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014
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