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Theorem frminex 4809
 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 3766 . 2
2 frminex.1 . . . . 5
32rabex 4552 . . . 4
4 ssrab2 3546 . . . 4
5 fri 4791 . . . . . 6
6 frminex.2 . . . . . . . . 9
76ralrab 3228 . . . . . . . 8
87rexbii 2862 . . . . . . 7
9 breq2 4405 . . . . . . . . . . 11
109notbid 294 . . . . . . . . . 10
1110imbi2d 316 . . . . . . . . 9
1211ralbidv 2846 . . . . . . . 8
1312rexrab2 3234 . . . . . . 7
148, 13bitri 249 . . . . . 6
155, 14sylib 196 . . . . 5
1615an4s 822 . . . 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   wcel 1758   wne 2648  wral 2799  wrex 2800  crab 2803  cvv 3078   wss 3437  c0 3746   class class class wbr 4401   wfr 4785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-fr 4788 This theorem is referenced by: (None)
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