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Theorem gchina 9080
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 7817 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 8072 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 9075 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  om  C_  x
)
7 ssdomg 7563 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  InaccW  ->  om  ~<_  x )
9 sdomdomtr 7652 . . . . . . . . . . . . . . . 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 3098 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 simplll 759 . . . . . . . . . . . . . . 15  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> GCH  =  _V )
1816, 17syl5eleqr 2538 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
19 simprr 757 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
20 gchinf 9038 . . . . . . . . . . . . . 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 3098 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2538 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 9051 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1229 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 9073 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  InaccW  ->  ( card `  x )  =  x )
27 iscard 8359 . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7654 . . . . . . . . . . . . . . 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 2935 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2851 . . . . . . 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 1307 . . . . . 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 9067 . . . . . 6  |-  ( x  e.  InaccW  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 9068 . . . . . 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 9071 . . 3  |-  ( x  e.  Inacc  ->  x  e.  InaccW )
4846, 47impbid1 203 . 2  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  <->  x  e.  Inacc ) )
4948eqrdv 2440 1  |-  (GCH  =  _V  ->  InaccW  =  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   class class class wbr 4437   Oncon0 4868   ` cfv 5578   omcom 6685    ~<_ cdom 7516    ~< csdm 7517   Fincfn 7518   cardccrd 8319   cfccf 8321  GCHcgch 9001   InaccWcwina 9063   Inacccina 9064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-har 7987  df-wdom 7988  df-cnf 8082  df-card 8323  df-cf 8325  df-cda 8551  df-fin4 8670  df-gch 9002  df-wina 9065  df-ina 9066
This theorem is referenced by: (None)
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