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Theorem gchina 8871
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 7611 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 7863 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 8866 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  om  C_  x
)
7 ssdomg 7360 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  InaccW  ->  om  ~<_  x )
9 sdomdomtr 7449 . . . . . . . . . . . . . . . 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 2980 . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 8829 . . . . . . . . . . . . . 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 2980 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2530 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 8842 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 8864 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  InaccW  ->  ( card `  x )  =  x )
27 iscard 8150 . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7451 . . . . . . . . . . . . . . 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 2846 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2799 . . . . . . 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 1296 . . . . . 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 8858 . . . . . 6  |-  ( x  e.  InaccW  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 8859 . . . . . 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 8862 . . 3  |-  ( x  e.  Inacc  ->  x  e.  InaccW )
4846, 47impbid1 203 . 2  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  <->  x  e.  Inacc ) )
4948eqrdv 2441 1  |-  (GCH  =  _V  ->  InaccW  =  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   class class class wbr 4297   Oncon0 4724   ` cfv 5423   omcom 6481    ~<_ cdom 7313    ~< csdm 7314   Fincfn 7315   cardccrd 8110   cfccf 8112  GCHcgch 8792   InaccWcwina 8854   Inacccina 8855
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-oexp 6931  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-har 7778  df-wdom 7779  df-cnf 7873  df-card 8114  df-cf 8116  df-cda 8342  df-fin4 8461  df-gch 8793  df-wina 8856  df-ina 8857
This theorem is referenced by: (None)
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