Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  onminex Structured version   Unicode version

Theorem onminex 4788
 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:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem onminex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3429 . . . 4
2 rabn0 3648 . . . . 5
32biimpri 199 . . . 4
4 oninton 4781 . . . 4
51, 3, 4sylancr 646 . . 3
6 onminesb 4779 . . 3
7 onss 4772 . . . . . . 7
85, 7syl 16 . . . . . 6
98sseld 3348 . . . . 5
10 onminex.1 . . . . . 6
1110onnminsb 4785 . . . . 5
129, 11syli 36 . . . 4
1312ralrimiv 2789 . . 3
14 dfsbcq2 3165 . . . . 5
15 raleq 2905 . . . . 5
1614, 15anbi12d 693 . . . 4
1716rspcev 3053 . . 3
185, 6, 13, 17syl12anc 1183 . 2
19 nfv 1630 . . 3
20 nfs1v 2183 . . . 4
21 nfv 1630 . . . 4
2220, 21nfan 1847 . . 3
23 sbequ12 1945 . . . 4
24 raleq 2905 . . . 4
2523, 24anbi12d 693 . . 3
2619, 22, 25cbvrex 2930 . 2
2718, 26sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653  wsb 1659   wcel 1726   wne 2600  wral 2706  wrex 2707  crab 2710  wsbc 3162   wss 3321  c0 3629  cint 4051  con0 4582 This theorem is referenced by:  tz7.49  6703  omeulem1  6826  zorn2lem7  8383 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-13 1728  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  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586
 Copyright terms: Public domain W3C validator