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Theorem exmidomniim 6925
 Description: Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 6926. (Contributed by Jim Kingdon, 29-Jun-2022.)
Assertion
Ref Expression
exmidomniim EXMID Omni

Proof of Theorem exmidomniim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exmidexmid 4060 . . . . . . . . 9 EXMID DECID
2 exmiddc 788 . . . . . . . . 9 DECID
31, 2syl 14 . . . . . . . 8 EXMID
43orcomd 689 . . . . . . 7 EXMID
54adantr 272 . . . . . 6 EXMID
6 ffvelrn 5485 . . . . . . . . . . . . . 14
7 df2o3 6257 . . . . . . . . . . . . . 14
86, 7syl6eleq 2192 . . . . . . . . . . . . 13
9 elpri 3497 . . . . . . . . . . . . 13
108, 9syl 14 . . . . . . . . . . . 12
1110ord 684 . . . . . . . . . . 11
1211ralimdva 2458 . . . . . . . . . 10
1312con3d 601 . . . . . . . . 9
1413adantl 273 . . . . . . . 8 EXMID
15 exmidexmid 4060 . . . . . . . . . 10 EXMID DECID
16 dfrex2dc 2387 . . . . . . . . . 10 DECID
1715, 16syl 14 . . . . . . . . 9 EXMID
1817adantr 272 . . . . . . . 8 EXMID
1914, 18sylibrd 168 . . . . . . 7 EXMID
2019orim1d 742 . . . . . 6 EXMID
215, 20mpd 13 . . . . 5 EXMID
2221ex 114 . . . 4 EXMID
2322alrimiv 1813 . . 3 EXMID
24 vex 2644 . . . 4
25 isomni 6920 . . . 4 Omni
2624, 25ax-mp 7 . . 3 Omni
2723, 26sylibr 133 . 2 EXMID Omni
2827alrimiv 1813 1 EXMID Omni
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 670  DECID wdc 786  wal 1297   wceq 1299   wcel 1448  wral 2375  wrex 2376  cvv 2641  c0 3310  cpr 3475  EXMIDwem 4058  wf 5055  cfv 5059  c1o 6236  c2o 6237  Omnicomni 6916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069 This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-exmid 4059  df-id 4153  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-1o 6243  df-2o 6244  df-omni 6918 This theorem is referenced by:  exmidomni  6926
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