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Theorem frminex 4389
 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 3487 . 2
2 frminex.1 . . . . 5
32rabex 4181 . . . 4
4 ssrab2 3271 . . . 4
5 fri 4371 . . . . . 6
6 frminex.2 . . . . . . . . 9
76ralrab 2940 . . . . . . . 8
87rexbii 2581 . . . . . . 7
9 breq2 4043 . . . . . . . . . . 11
109notbid 285 . . . . . . . . . 10
1110imbi2d 307 . . . . . . . . 9
1211ralbidv 2576 . . . . . . . 8
1312rexrab2 2946 . . . . . . 7
148, 13bitri 240 . . . . . 6
155, 14sylib 188 . . . . 5
1615an4s 799 . . . 4
173, 4, 16mpanl12 663 . . 3
1817ex 423 . 2
191, 18syl5bir 209 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1632   wcel 1696   wne 2459  wral 2556  wrex 2557  crab 2560  cvv 2801   wss 3165  c0 3468   class class class wbr 4039   wfr 4365 This theorem is referenced by:  frminexOLD  26532 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-fr 4368
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