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Theorem frminex 4562
Description: If an element of a well-founded set satisfies a property  ph, then there is a minimal element that satisfies  ph. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
frminex.1  |-  A  e. 
_V
frminex.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
frminex  |-  ( R  Fr  A  ->  ( E. x  e.  A  ph 
->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
Distinct variable groups:    x, A, y    x, R, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem frminex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rabn0 3647 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
2 frminex.1 . . . . 5  |-  A  e. 
_V
32rabex 4354 . . . 4  |-  { x  e.  A  |  ph }  e.  _V
4 ssrab2 3428 . . . 4  |-  { x  e.  A  |  ph }  C_  A
5 fri 4544 . . . . . 6  |-  ( ( ( { x  e.  A  |  ph }  e.  _V  /\  R  Fr  A )  /\  ( { x  e.  A  |  ph }  C_  A  /\  { x  e.  A  |  ph }  =/=  (/) ) )  ->  E. z  e.  {
x  e.  A  |  ph } A. y  e. 
{ x  e.  A  |  ph }  -.  y R z )
6 frminex.2 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76ralrab 3096 . . . . . . . 8  |-  ( A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  A. y  e.  A  ( ps  ->  -.  y R z ) )
87rexbii 2730 . . . . . . 7  |-  ( E. z  e.  { x  e.  A  |  ph } A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  E. z  e.  { x  e.  A  |  ph } A. y  e.  A  ( ps  ->  -.  y R z ) )
9 breq2 4216 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
y R z  <->  y R x ) )
109notbid 286 . . . . . . . . . 10  |-  ( z  =  x  ->  ( -.  y R z  <->  -.  y R x ) )
1110imbi2d 308 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ps  ->  -.  y R z )  <->  ( ps  ->  -.  y R x ) ) )
1211ralbidv 2725 . . . . . . . 8  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  -.  y R z )  <->  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1312rexrab2 3102 . . . . . . 7  |-  ( E. z  e.  { x  e.  A  |  ph } A. y  e.  A  ( ps  ->  -.  y R z )  <->  E. x  e.  A  ( ph  /\ 
A. y  e.  A  ( ps  ->  -.  y R x ) ) )
148, 13bitri 241 . . . . . 6  |-  ( E. z  e.  { x  e.  A  |  ph } A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  E. x  e.  A  ( ph  /\ 
A. y  e.  A  ( ps  ->  -.  y R x ) ) )
155, 14sylib 189 . . . . 5  |-  ( ( ( { x  e.  A  |  ph }  e.  _V  /\  R  Fr  A )  /\  ( { x  e.  A  |  ph }  C_  A  /\  { x  e.  A  |  ph }  =/=  (/) ) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1615an4s 800 . . . 4  |-  ( ( ( { x  e.  A  |  ph }  e.  _V  /\  { x  e.  A  |  ph }  C_  A )  /\  ( R  Fr  A  /\  { x  e.  A  |  ph }  =/=  (/) ) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
173, 4, 16mpanl12 664 . . 3  |-  ( ( R  Fr  A  /\  { x  e.  A  |  ph }  =/=  (/) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1817ex 424 . 2  |-  ( R  Fr  A  ->  ( { x  e.  A  |  ph }  =/=  (/)  ->  E. x  e.  A  ( ph  /\ 
A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
191, 18syl5bir 210 1  |-  ( R  Fr  A  ->  ( E. x  e.  A  ph 
->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   class class class wbr 4212    Fr 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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