MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frminex Unicode version

Theorem frminex 4389
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 3487 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
2 frminex.1 . . . . 5  |-  A  e. 
_V
32rabex 4181 . . . 4  |-  { x  e.  A  |  ph }  e.  _V
4 ssrab2 3271 . . . 4  |-  { x  e.  A  |  ph }  C_  A
5 fri 4371 . . . . . 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 2940 . . . . . . . 8  |-  ( A. y  e.  { x  e.  A  |  ph }  -.  y R z  <->  A. y  e.  A  ( ps  ->  -.  y R z ) )
87rexbii 2581 . . . . . . 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 4043 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
y R z  <->  y R x ) )
109notbid 285 . . . . . . . . . 10  |-  ( z  =  x  ->  ( -.  y R z  <->  -.  y R x ) )
1110imbi2d 307 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ps  ->  -.  y R z )  <->  ( ps  ->  -.  y R x ) ) )
1211ralbidv 2576 . . . . . . . 8  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  -.  y R z )  <->  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1312rexrab2 2946 . . . . . . 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 240 . . . . . 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 188 . . . . 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 799 . . . 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 663 . . 3  |-  ( ( R  Fr  A  /\  { x  e.  A  |  ph }  =/=  (/) )  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) )
1817ex 423 . 2  |-  ( R  Fr  A  ->  ( { x  e.  A  |  ph }  =/=  (/)  ->  E. x  e.  A  ( ph  /\ 
A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
191, 18syl5bir 209 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039    Fr 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
  Copyright terms: Public domain W3C validator