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

Theorem frminex 4864
Description: If an element of a well-founded set satisfies a property , then there is a minimal element that satisfies . (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
frminex.1
frminex.2
Assertion
Ref Expression
frminex
Distinct variable groups:   , ,   , ,   ,   ,

Proof of Theorem frminex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rabn0 3805 . 2
2 frminex.1 . . . . 5
32rabex 4603 . . . 4
4 ssrab2 3584 . . . 4
5 fri 4846 . . . . . 6
6 frminex.2 . . . . . . . . 9
76ralrab 3261 . . . . . . . 8
87rexbii 2959 . . . . . . 7
9 breq2 4456 . . . . . . . . . . 11
109notbid 294 . . . . . . . . . 10
1110imbi2d 316 . . . . . . . . 9
1211ralbidv 2896 . . . . . . . 8
1312rexrab2 3267 . . . . . . 7
148, 13bitri 249 . . . . . 6
155, 14sylib 196 . . . . 5
1615an4s 826 . . . 4
173, 4, 16mpanl12 682 . . 3
1817ex 434 . 2
191, 18syl5bir 218 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  {crab 2811   cvv 3109  C_wss 3475   c0 3784   class class class wbr 4452  Frwfr 4840
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573
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-ral 2812  df-rex 2813  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-fr 4843
  Copyright terms: Public domain W3C validator