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Theorem gchina 9142
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 468 . . . . 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 7887 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 8175 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 9137 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  om  C_  x
)
7 ssdomg 7633 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  InaccW  ->  om  ~<_  x )
9 sdomdomtr 7723 . . . . . . . . . . . . . . . 16  |-  ( ( ~P y  ~<  om  /\  om  ~<_  x )  ->  ~P y  ~<  x )
109expcom 442 . . . . . . . . . . . . . . 15  |-  ( om  ~<_  x  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x ) )
118, 10syl 17 . . . . . . . . . . . . . 14  |-  ( x  e.  InaccW  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x )
)
125, 11syl5bi 225 . . . . . . . . . . . . 13  |-  ( x  e.  InaccW  ->  ( ~P y  e.  Fin  ->  ~P y  ~<  x
) )
134, 12syl5bi 225 . . . . . . . . . . . 12  |-  ( x  e.  InaccW  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1413ad3antlr 745 . . . . . . . . . . 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 3034 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 simplll 776 . . . . . . . . . . . . . . 15  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> GCH  =  _V )
1816, 17syl5eleqr 2556 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
19 simprr 774 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
20 gchinf 9100 . . . . . . . . . . . . . 14  |-  ( ( y  e. GCH  /\  -.  y  e.  Fin )  ->  om  ~<_  y )
2118, 19, 20syl2anc 673 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  om 
~<_  y )
22 vex 3034 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2556 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 9113 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1292 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 9135 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  InaccW  ->  ( card `  x )  =  x )
27 iscard 8427 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. z  e.  x  z  ~<  x ) )
2827simprbi 471 . . . . . . . . . . . . . . . . 17  |-  ( (
card `  x )  =  x  ->  A. z  e.  x  z  ~<  x )
2926, 28syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  A. z  e.  x  z  ~<  x )
3029ad2antlr 741 . . . . . . . . . . . . . . 15  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  A. z  e.  x  z  ~<  x )
3130r19.21bi 2776 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7725 . . . . . . . . . . . . . . 15  |-  ( ( ~P y  ~<_  z  /\  z  ~<  x )  ->  ~P y  ~<  x )
3332expcom 442 . . . . . . . . . . . . . 14  |-  ( z 
~<  x  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3431, 33syl 17 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x )
)
3534adantrr 731 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3625, 35sylbid 223 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  ->  ~P y  ~<  x
) )
3736expr 626 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( -.  y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x
) ) )
3815, 37pm2.61d 163 . . . . . . . . 9  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  ~<  z  ->  ~P y  ~<  x ) )
3938rexlimdva 2871 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2805 . . . . . . 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 1372 . . . . . 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 9129 . . . . . 6  |-  ( x  e.  InaccW  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 9130 . . . . . 6  |-  ( x  e.  Inacc 
<->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) )
4441, 42, 433imtr4g 278 . . . . 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 441 . . 3  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  ->  x  e. 
Inacc ) )
47 inawina 9133 . . 3  |-  ( x  e.  Inacc  ->  x  e.  InaccW )
4846, 47impbid1 208 . 2  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  <->  x  e.  Inacc ) )
4948eqrdv 2469 1  |-  (GCH  =  _V  ->  InaccW  =  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   class class class wbr 4395   Oncon0 5430   ` cfv 5589   omcom 6711    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587   cardccrd 8387   cfccf 8389  GCHcgch 9063   InaccWcwina 9125   Inacccina 9126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-oexp 7206  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-har 8091  df-wdom 8092  df-cnf 8185  df-card 8391  df-cf 8393  df-cda 8616  df-fin4 8735  df-gch 9064  df-wina 9127  df-ina 9128
This theorem is referenced by: (None)
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