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Theorem wsuclem 28986
Description: Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.)
Hypotheses
Ref Expression
wsuclem.1  |-  ( ph  ->  R  We  A )
wsuclem.2  |-  ( ph  ->  R Se  A )
wsuclem.3  |-  ( ph  ->  X  e.  V )
wsuclem.4  |-  ( ph  ->  E. w  e.  A  X R w )
Assertion
Ref Expression
wsuclem  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) )
Distinct variable groups:    x, A, y, z, w    ph, x, y    x, R, y, z, w    x, X, y, z, w
Allowed substitution hints:    ph( z, w)    V( x, y, z, w)

Proof of Theorem wsuclem
StepHypRef Expression
1 wsuclem.1 . . 3  |-  ( ph  ->  R  We  A )
2 wsuclem.2 . . 3  |-  ( ph  ->  R Se  A )
3 predss 28856 . . . 4  |-  Pred ( `' R ,  A ,  X )  C_  A
43a1i 11 . . 3  |-  ( ph  ->  Pred ( `' R ,  A ,  X ) 
C_  A )
5 wsuclem.3 . . . . 5  |-  ( ph  ->  X  e.  V )
6 dfpred3g 28860 . . . . 5  |-  ( X  e.  V  ->  Pred ( `' R ,  A ,  X )  =  {
w  e.  A  |  w `' R X } )
75, 6syl 16 . . . 4  |-  ( ph  ->  Pred ( `' R ,  A ,  X )  =  { w  e.  A  |  w `' R X } )
8 elex 3122 . . . . . 6  |-  ( X  e.  V  ->  X  e.  _V )
95, 8syl 16 . . . . 5  |-  ( ph  ->  X  e.  _V )
10 wsuclem.4 . . . . 5  |-  ( ph  ->  E. w  e.  A  X R w )
11 rabn0 3805 . . . . . . 7  |-  ( { w  e.  A  |  w `' R X }  =/=  (/)  <->  E. w  e.  A  w `' R X )
12 brcnvg 5183 . . . . . . . . 9  |-  ( ( w  e.  A  /\  X  e.  _V )  ->  ( w `' R X 
<->  X R w ) )
1312ancoms 453 . . . . . . . 8  |-  ( ( X  e.  _V  /\  w  e.  A )  ->  ( w `' R X 
<->  X R w ) )
1413rexbidva 2970 . . . . . . 7  |-  ( X  e.  _V  ->  ( E. w  e.  A  w `' R X  <->  E. w  e.  A  X R w ) )
1511, 14syl5bb 257 . . . . . 6  |-  ( X  e.  _V  ->  ( { w  e.  A  |  w `' R X }  =/=  (/)  <->  E. w  e.  A  X R w ) )
1615biimpar 485 . . . . 5  |-  ( ( X  e.  _V  /\  E. w  e.  A  X R w )  ->  { w  e.  A  |  w `' R X }  =/=  (/) )
179, 10, 16syl2anc 661 . . . 4  |-  ( ph  ->  { w  e.  A  |  w `' R X }  =/=  (/) )
187, 17eqnetrd 2760 . . 3  |-  ( ph  ->  Pred ( `' R ,  A ,  X )  =/=  (/) )
19 tz6.26 28890 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Pred ( `' R ,  A ,  X )  C_  A  /\  Pred ( `' R ,  A ,  X )  =/=  (/) ) )  ->  E. x  e.  Pred  ( `' R ,  A ,  X ) Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )
201, 2, 4, 18, 19syl22anc 1229 . 2  |-  ( ph  ->  E. x  e.  Pred  ( `' R ,  A ,  X ) Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )
21 dfpred3g 28860 . . . . 5  |-  ( X  e.  V  ->  Pred ( `' R ,  A ,  X )  =  {
y  e.  A  | 
y `' R X } )
225, 21syl 16 . . . 4  |-  ( ph  ->  Pred ( `' R ,  A ,  X )  =  { y  e.  A  |  y `' R X } )
2322rexeqdv 3065 . . 3  |-  ( ph  ->  ( E. x  e. 
Pred  ( `' R ,  A ,  X )
Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x )  =  (/)  <->  E. x  e.  { y  e.  A  |  y `' R X } Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) ) )
24 breq1 4450 . . . . 5  |-  ( y  =  x  ->  (
y `' R X  <-> 
x `' R X ) )
2524rexrab 3267 . . . 4  |-  ( E. x  e.  { y  e.  A  |  y `' R X } Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/)  <->  E. x  e.  A  ( x `' R X  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) ) )
26 noel 3789 . . . . . . . . . . . . 13  |-  -.  y  e.  (/)
27 simp2r 1023 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )
2827eleq2d 2537 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  -> 
( y  e.  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  <->  y  e.  (/) ) )
2926, 28mtbiri 303 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  -.  y  e.  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
) )
30 vex 3116 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3130a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  x  e.  _V )
32 simp3 998 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  -> 
y  e.  Pred ( `' R ,  A ,  X ) )
33 elpredg 28863 . . . . . . . . . . . . 13  |-  ( ( x  e.  _V  /\  y  e.  Pred ( `' R ,  A ,  X ) )  -> 
( y  e.  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  <->  y R x ) )
3431, 32, 33syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  -> 
( y  e.  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  <->  y R x ) )
3529, 34mtbid 300 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  -.  y R x )
36 vex 3116 . . . . . . . . . . . 12  |-  y  e. 
_V
3730, 36brcnv 5185 . . . . . . . . . . 11  |-  ( x `' R y  <->  y R x )
3835, 37sylnibr 305 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  -.  x `' R y )
39383expa 1196 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) ) )  /\  y  e.  Pred ( `' R ,  A ,  X ) )  ->  -.  x `' R y )
4039ralrimiva 2878 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) ) )  ->  A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y )
4140expr 615 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/)  ->  A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R
y ) )
42 simp1rl 1061 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  x  e.  A )
43 simp1rr 1062 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  x `' R X )
44 simp1l 1020 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  ph )
4544, 5syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  X  e.  V )
4630elpred 28862 . . . . . . . . . . . . 13  |-  ( X  e.  V  ->  (
x  e.  Pred ( `' R ,  A ,  X )  <->  ( x  e.  A  /\  x `' R X ) ) )
4745, 46syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  ( x  e.  Pred ( `' R ,  A ,  X )  <-> 
( x  e.  A  /\  x `' R X ) ) )
4842, 43, 47mpbir2and 920 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  x  e.  Pred ( `' R ,  A ,  X )
)
49 simp3 998 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  y `' R x )
50 breq2 4451 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
y `' R z  <-> 
y `' R x ) )
5150rspcev 3214 . . . . . . . . . . 11  |-  ( ( x  e.  Pred ( `' R ,  A ,  X )  /\  y `' R x )  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z )
5248, 49, 51syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A  /\  y `' R x )  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z )
53523expia 1198 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  x `' R X ) )  /\  y  e.  A
)  ->  ( y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) )
5453ralrimiva 2878 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  x `' R X ) )  ->  A. y  e.  A  ( y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) )
5554expr 615 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
x `' R X  ->  A. y  e.  A  ( y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) )
5641, 55anim12d 563 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x )  =  (/)  /\  x `' R X )  ->  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) ) )
5756ancomsd 454 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x `' R X  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x )  =  (/) )  ->  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R
y  /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) ) )
5857reximdva 2938 . . . 4  |-  ( ph  ->  ( E. x  e.  A  ( x `' R X  /\  Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x
)  =  (/) )  ->  E. x  e.  A  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) ) )
5925, 58syl5bi 217 . . 3  |-  ( ph  ->  ( E. x  e. 
{ y  e.  A  |  y `' R X } Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x )  =  (/)  ->  E. x  e.  A  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) ) )
6023, 59sylbid 215 . 2  |-  ( ph  ->  ( E. x  e. 
Pred  ( `' R ,  A ,  X )
Pred ( R ,  Pred ( `' R ,  A ,  X ) ,  x )  =  (/)  ->  E. x  e.  A  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) ) )
6120, 60mpd 15 1  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
y `' R x  ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447   Se wse 4836    We wwe 4837   `'ccnv 4998   Predcpred 28848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 28849
This theorem is referenced by:  wsuccl  28988  wsuclb  28989
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