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Theorem dmsnopss 5343
 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 5342 . . 3
2 eqimss 3401 . . 3
31, 2syl 16 . 2
4 opprc2 4008 . . . . . 6
54sneqd 3828 . . . . 5
65dmeqd 5073 . . . 4
7 dmsn0 5338 . . . 4
86, 7syl6eq 2485 . . 3
9 0ss 3657 . . 3
108, 9syl6eqss 3399 . 2
113, 10pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1653   wcel 1726  cvv 2957   wss 3321  c0 3629  csn 3815  cop 3818   cdm 4879 This theorem is referenced by:  setsres  13496  setscom  13498  setsid  13509  strlemor1  13557  strle1  13561  constr3pthlem1  21643  ex-res  21750  funsnfsup  26744  mapfzcons1  26774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-dm 4889
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