MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn2lem3 Structured version   Unicode version

Theorem zorn2lem3 8665
Description: Lemma for zorn2 8673. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
zorn2lem.3  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
zorn2lem.4  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
zorn2lem.5  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
Assertion
Ref Expression
zorn2lem3  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
Distinct variable groups:    f, g, u, v, w, x, y, z, A    D, f, u, v, y    f, F, g, u, v, x, y, z    R, f, g, u, v, w, x, y, z    v, C
Allowed substitution hints:    C( x, y, z, w, u, f, g)    D( x, z, w, g)    F( w)

Proof of Theorem zorn2lem3
StepHypRef Expression
1 zorn2lem.3 . . . 4  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
2 zorn2lem.4 . . . 4  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
3 zorn2lem.5 . . . 4  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
41, 2, 3zorn2lem2 8664 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( y  e.  x  ->  ( F `  y ) R ( F `  x ) ) )
54adantl 466 . 2  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  ( F `
 y ) R ( F `  x
) ) )
6 ssrab2 3435 . . . . 5  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
73, 6eqsstri 3384 . . . 4  |-  D  C_  A
81, 2, 3zorn2lem1 8663 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
97, 8sseldi 3352 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
10 breq1 4293 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  (
( F `  x
) R ( F `
 x )  <->  ( F `  y ) R ( F `  x ) ) )
1110biimprcd 225 . . . . 5  |-  ( ( F `  y ) R ( F `  x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
) R ( F `
 x ) ) )
12 poirr 4650 . . . . 5  |-  ( ( R  Po  A  /\  ( F `  x )  e.  A )  ->  -.  ( F `  x
) R ( F `
 x ) )
1311, 12nsyli 141 . . . 4  |-  ( ( F `  y ) R ( F `  x )  ->  (
( R  Po  A  /\  ( F `  x
)  e.  A )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1413com12 31 . . 3  |-  ( ( R  Po  A  /\  ( F `  x )  e.  A )  -> 
( ( F `  y ) R ( F `  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
159, 14sylan2 474 . 2  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( ( F `  y ) R ( F `  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
165, 15syld 44 1  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970   (/)c0 3635   class class class wbr 4290    e. cmpt 4348    Po wpo 4637    We wwe 4676   Oncon0 4717   ran crn 4839   "cima 4841   ` cfv 5416   iota_crio 6049  recscrecs 6829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-recs 6830
This theorem is referenced by:  zorn2lem4  8666
  Copyright terms: Public domain W3C validator