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Theorem frminex 4809
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 3766 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
2 frminex.1 . . . . 5  |-  A  e. 
_V
32rabex 4552 . . . 4  |-  { x  e.  A  |  ph }  e.  _V
4 ssrab2 3546 . . . 4  |-  { x  e.  A  |  ph }  C_  A
5 fri 4791 . . . . . 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 3228 . . . . . . . 8  |-  ( A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  A. y  e.  A  ( ps  ->  -.  y R z ) )
87rexbii 2862 . . . . . . 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 4405 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
y R z  <->  y R x ) )
109notbid 294 . . . . . . . . . 10  |-  ( z  =  x  ->  ( -.  y R z  <->  -.  y R x ) )
1110imbi2d 316 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ps  ->  -.  y R z )  <->  ( ps  ->  -.  y R x ) ) )
1211ralbidv 2846 . . . . . . . 8  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  -.  y R z )  <->  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1312rexrab2 3234 . . . . . . 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 822 . . . 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 682 . . 3  |-  ( ( R  Fr  A  /\  { x  e.  A  |  ph }  =/=  (/) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1817ex 434 . 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 369    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078    C_ wss 3437   (/)c0 3746   class class class wbr 4401    Fr 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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