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Theorem gchina 8321
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  ->  Inacc W  =  Inacc )

Proof of Theorem gchina
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  ->  x  e.  Inacc W )
2 idd 21 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( x  =/=  (/)  ->  x  =/=  (/) ) )
3 idd 21 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( ( cf `  x
)  =  x  -> 
( cf `  x
)  =  x ) )
4 simpllr 735 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  x  e.  Inacc W )
5 pwfi 7151 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
6 isfinite 7353 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
7 winainf 8316 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  om  C_  x
)
8 ssdomg 6907 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  ( om  C_  x  ->  om  ~<_  x ) )
97, 8mpd 14 . . . . . . . . . . . . . . 15  |-  ( x  e.  Inacc W  ->  om  ~<_  x )
10 sdomdomtr 6994 . . . . . . . . . . . . . . . 16  |-  ( ( ~P y  ~<  om  /\  om  ~<_  x )  ->  ~P y  ~<  x )
1110expcom 424 . . . . . . . . . . . . . . 15  |-  ( om  ~<_  x  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x ) )
129, 11syl 15 . . . . . . . . . . . . . 14  |-  ( x  e.  Inacc W  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x )
)
136, 12syl5bi 208 . . . . . . . . . . . . 13  |-  ( x  e.  Inacc W  ->  ( ~P y  e.  Fin  ->  ~P y  ~<  x
) )
145, 13syl5bi 208 . . . . . . . . . . . 12  |-  ( x  e.  Inacc W  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
154, 14syl 15 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1615a1dd 42 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x )
) )
17 vex 2791 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
18 simplll 734 . . . . . . . . . . . . . . 15  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> GCH  =  _V )
1917, 18syl5eleqr 2370 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
20 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
21 gchinf 8279 . . . . . . . . . . . . . 14  |-  ( ( y  e. GCH  /\  -.  y  e.  Fin )  ->  om  ~<_  y )
2219, 20, 21syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  om 
~<_  y )
23 vex 2791 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2423, 18syl5eleqr 2370 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
25 gchpwdom 8296 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2622, 19, 24, 25syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
27 winacard 8314 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  Inacc W  ->  ( card `  x )  =  x )
28 iscard 7608 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. z  e.  x  z  ~<  x ) )
2928simprbi 450 . . . . . . . . . . . . . . . . 17  |-  ( (
card `  x )  =  x  ->  A. z  e.  x  z  ~<  x )
3027, 29syl 15 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  A. z  e.  x  z  ~<  x )
3130ad2antlr 707 . . . . . . . . . . . . . . 15  |-  ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  ->  A. z  e.  x  z  ~<  x )
3231r19.21bi 2641 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
33 domsdomtr 6996 . . . . . . . . . . . . . . 15  |-  ( ( ~P y  ~<_  z  /\  z  ~<  x )  ->  ~P y  ~<  x )
3433expcom 424 . . . . . . . . . . . . . 14  |-  ( z 
~<  x  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3532, 34syl 15 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x )
)
3635adantrr 697 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3726, 36sylbid 206 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  ->  ~P y  ~<  x
) )
3837expr 598 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( -.  y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x
) ) )
3916, 38pm2.61d 150 . . . . . . . . 9  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  ~<  z  ->  ~P y  ~<  x ) )
4039rexlimdva 2667 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4140ralimdva 2621 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( A. y  e.  x  E. z  e.  x  y  ~<  z  ->  A. y  e.  x  ~P y  ~<  x ) )
422, 3, 413anim123d 1259 . . . . . 6  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( ( 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 ) ) )
43 elwina 8308 . . . . . 6  |-  ( x  e.  Inacc W  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
44 elina 8309 . . . . . 6  |-  ( x  e.  Inacc 
<->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) )
4542, 43, 443imtr4g 261 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( x  e.  Inacc W  ->  x  e.  Inacc ) )
461, 45mpd 14 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  ->  x  e.  Inacc )
4746ex 423 . . 3  |-  (GCH  =  _V  ->  ( x  e. 
Inacc W  ->  x  e. 
Inacc ) )
48 inawina 8312 . . 3  |-  ( x  e.  Inacc  ->  x  e.  Inacc W )
4947, 48impbid1 194 . 2  |-  (GCH  =  _V  ->  ( x  e. 
Inacc W  <->  x  e.  Inacc ) )
5049eqrdv 2281 1  |-  (GCH  =  _V  ->  Inacc W  =  Inacc )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   omcom 4656   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   cardccrd 7568   cfccf 7570  GCHcgch 8242   Inacc Wcwina 8304   Inacccina 8305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-wdom 7273  df-cnf 7363  df-card 7572  df-cf 7574  df-cda 7794  df-fin4 7913  df-gch 8243  df-wina 8306  df-ina 8307
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