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Theorem onminex 6642
Description: If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)
Hypothesis
Ref Expression
onminex.1
Assertion
Ref Expression
onminex
Distinct variable groups:   ,   ,   ,

Proof of Theorem onminex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3584 . . . 4
2 rabn0 3805 . . . . 5
32biimpri 206 . . . 4
4 oninton 6635 . . . 4
51, 3, 4sylancr 663 . . 3
6 onminesb 6633 . . 3
7 onss 6626 . . . . . . 7
85, 7syl 16 . . . . . 6
98sseld 3502 . . . . 5
10 onminex.1 . . . . . 6
1110onnminsb 6639 . . . . 5
129, 11syli 37 . . . 4
1312ralrimiv 2869 . . 3
14 dfsbcq2 3330 . . . . 5
15 raleq 3054 . . . . 5
1614, 15anbi12d 710 . . . 4
1716rspcev 3210 . . 3
185, 6, 13, 17syl12anc 1226 . 2
19 nfv 1707 . . 3
20 nfs1v 2181 . . . 4
21 nfv 1707 . . . 4
2220, 21nfan 1928 . . 3
23 sbequ12 1992 . . . 4
24 raleq 3054 . . . 4
2523, 24anbi12d 710 . . 3
2619, 22, 25cbvrex 3081 . 2
2718, 26sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  {crab 2811  [.wsbc 3327  C_wss 3475   c0 3784  |^|cint 4286   con0 4883
This theorem is referenced by:  tz7.49  7129  omeulem1  7250  zorn2lem7  8903
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-8 1820  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  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887
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