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Theorem dmsnn0 5415
 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 3081 . . . . 5
21eldm 5148 . . . 4
3 df-br 4404 . . . . . 6
4 opex 4667 . . . . . . 7
54elsnc 4012 . . . . . 6
6 eqcom 2463 . . . . . 6
73, 5, 63bitri 271 . . . . 5
87exbii 1635 . . . 4
92, 8bitr2i 250 . . 3
109exbii 1635 . 2
11 elvv 5008 . 2
12 n0 3757 . 2
1310, 11, 123bitr4i 277 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1370  wex 1587   wcel 1758   wne 2648  cvv 3078  c0 3748  csn 3988  cop 3994   class class class wbr 4403   cxp 4949   cdm 4951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-dm 4961 This theorem is referenced by:  rnsnn0  5416  dmsn0  5417  dmsn0el  5419  relsn2  5420  1stnpr  6694  1st2val  6715  mpt2xopxnop0  6845  hashfun  12321
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