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Theorem frminex 4562
 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:   ,,   ,,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem frminex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rabn0 3647 . 2
2 frminex.1 . . . . 5
32rabex 4354 . . . 4
4 ssrab2 3428 . . . 4
5 fri 4544 . . . . . 6
6 frminex.2 . . . . . . . . 9
76ralrab 3096 . . . . . . . 8
87rexbii 2730 . . . . . . 7
9 breq2 4216 . . . . . . . . . . 11
109notbid 286 . . . . . . . . . 10
1110imbi2d 308 . . . . . . . . 9
1211ralbidv 2725 . . . . . . . 8
1312rexrab2 3102 . . . . . . 7
148, 13bitri 241 . . . . . 6
155, 14sylib 189 . . . . 5
1615an4s 800 . . . 4
173, 4, 16mpanl12 664 . . 3
1817ex 424 . 2
191, 18syl5bir 210 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wcel 1725   wne 2599  wral 2705  wrex 2706  crab 2709  cvv 2956   wss 3320  c0 3628   class class class wbr 4212   wfr 4538 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-fr 4541
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