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Theorem dmsnopss 5485
Description: The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on ). (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
dmsnopss

Proof of Theorem dmsnopss
StepHypRef Expression
1 dmsnopg 5484 . . 3
2 eqimss 3555 . . 3
31, 2syl 16 . 2
4 opprc2 4241 . . . . . 6
54sneqd 4041 . . . . 5
65dmeqd 5210 . . . 4
7 dmsn0 5480 . . . 4
86, 7syl6eq 2514 . . 3
9 0ss 3814 . . 3
108, 9syl6eqss 3553 . 2
113, 10pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  =wceq 1395  e.wcel 1818   cvv 3109  C_wss 3475   c0 3784  {csn 4029  <.cop 4035  domcdm 5004
This theorem is referenced by:  snopsuppss  6933  setsres  14660  setscom  14662  setsid  14673  strlemor1  14724  strle1  14728  funsnfsupOLD  18256  constr3pthlem1  24655  ex-res  25162  mapfzcons1  30649
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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