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Theorem frminex 4773
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 3732 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
2 frminex.1 . . . . 5  |-  A  e. 
_V
32rabex 4516 . . . 4  |-  { x  e.  A  |  ph }  e.  _V
4 ssrab2 3499 . . . 4  |-  { x  e.  A  |  ph }  C_  A
5 fri 4755 . . . . . 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 3186 . . . . . . . 8  |-  ( A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  A. y  e.  A  ( ps  ->  -.  y R z ) )
87rexbii 2884 . . . . . . 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 4371 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
y R z  <->  y R x ) )
109notbid 292 . . . . . . . . . 10  |-  ( z  =  x  ->  ( -.  y R z  <->  -.  y R x ) )
1110imbi2d 314 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ps  ->  -.  y R z )  <->  ( ps  ->  -.  y R x ) ) )
1211ralbidv 2821 . . . . . . . 8  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  -.  y R z )  <->  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1312rexrab2 3192 . . . . . . 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 249 . . . . . 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 196 . . . . 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 824 . . . 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 680 . . 3  |-  ( ( R  Fr  A  /\  { x  e.  A  |  ph }  =/=  (/) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1817ex 432 . 2  |-  ( R  Fr  A  ->  ( { x  e.  A  |  ph }  =/=  (/)  ->  E. x  e.  A  ( ph  /\ 
A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
191, 18syl5bir 218 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034    C_ wss 3389   (/)c0 3711   class class class wbr 4367    Fr wfr 4749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-fr 4752
This theorem is referenced by: (None)
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