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Theorem gchina 9089
Description: Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
gchina  |-  (GCH  =  _V  ->  InaccW  =  Inacc )

Proof of Theorem gchina
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  ->  x  e.  InaccW )
2 idd 24 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  -> 
( x  =/=  (/)  ->  x  =/=  (/) ) )
3 idd 24 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  -> 
( ( cf `  x
)  =  x  -> 
( cf `  x
)  =  x ) )
4 pwfi 7827 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 8081 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 9084 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  om  C_  x
)
7 ssdomg 7573 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  InaccW  ->  om  ~<_  x )
9 sdomdomtr 7662 . . . . . . . . . . . . . . . 16  |-  ( ( ~P y  ~<  om  /\  om  ~<_  x )  ->  ~P y  ~<  x )
109expcom 435 . . . . . . . . . . . . . . 15  |-  ( om  ~<_  x  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x ) )
118, 10syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  InaccW  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x )
)
125, 11syl5bi 217 . . . . . . . . . . . . 13  |-  ( x  e.  InaccW  ->  ( ~P y  e.  Fin  ->  ~P y  ~<  x
) )
134, 12syl5bi 217 . . . . . . . . . . . 12  |-  ( x  e.  InaccW  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1413ad3antlr 730 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1514a1dd 46 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x )
) )
16 vex 3121 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 simplll 757 . . . . . . . . . . . . . . 15  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> GCH  =  _V )
1816, 17syl5eleqr 2562 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
19 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
20 gchinf 9047 . . . . . . . . . . . . . 14  |-  ( ( y  e. GCH  /\  -.  y  e.  Fin )  ->  om  ~<_  y )
2118, 19, 20syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  om 
~<_  y )
22 vex 3121 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2562 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 9060 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1228 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 9082 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  InaccW  ->  ( card `  x )  =  x )
27 iscard 8368 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. z  e.  x  z  ~<  x ) )
2827simprbi 464 . . . . . . . . . . . . . . . . 17  |-  ( (
card `  x )  =  x  ->  A. z  e.  x  z  ~<  x )
2926, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  A. z  e.  x  z  ~<  x )
3029ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  A. z  e.  x  z  ~<  x )
3130r19.21bi 2836 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7664 . . . . . . . . . . . . . . 15  |-  ( ( ~P y  ~<_  z  /\  z  ~<  x )  ->  ~P y  ~<  x )
3332expcom 435 . . . . . . . . . . . . . 14  |-  ( z 
~<  x  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3431, 33syl 16 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x )
)
3534adantrr 716 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3625, 35sylbid 215 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  ->  ~P y  ~<  x
) )
3736expr 615 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( -.  y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x
) ) )
3815, 37pm2.61d 158 . . . . . . . . 9  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  ~<  z  ->  ~P y  ~<  x ) )
3938rexlimdva 2959 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2875 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  -> 
( A. y  e.  x  E. z  e.  x  y  ~<  z  ->  A. y  e.  x  ~P y  ~<  x ) )
412, 3, 403anim123d 1306 . . . . . 6  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  -> 
( ( x  =/=  (/)  /\  ( cf `  x
)  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z )  ->  (
x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) ) )
42 elwina 9076 . . . . . 6  |-  ( x  e.  InaccW  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 9077 . . . . . 6  |-  ( x  e.  Inacc 
<->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) )
4441, 42, 433imtr4g 270 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  -> 
( x  e.  InaccW  ->  x  e.  Inacc ) )
451, 44mpd 15 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  InaccW )  ->  x  e.  Inacc )
4645ex 434 . . 3  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  ->  x  e. 
Inacc ) )
47 inawina 9080 . . 3  |-  ( x  e.  Inacc  ->  x  e.  InaccW )
4846, 47impbid1 203 . 2  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  <->  x  e.  Inacc ) )
4948eqrdv 2464 1  |-  (GCH  =  _V  ->  InaccW  =  Inacc )
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 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   class class class wbr 4453   Oncon0 4884   ` cfv 5594   omcom 6695    ~<_ cdom 7526    ~< csdm 7527   Fincfn 7528   cardccrd 8328   cfccf 8330  GCHcgch 9010   InaccWcwina 9072   Inacccina 9073
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-oexp 7148  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-har 7996  df-wdom 7997  df-cnf 8091  df-card 8332  df-cf 8334  df-cda 8560  df-fin4 8679  df-gch 9011  df-wina 9074  df-ina 9075
This theorem is referenced by: (None)
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