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Theorem gchina 8534
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 448 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  ->  x  e.  Inacc W )
2 idd 22 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( x  =/=  (/)  ->  x  =/=  (/) ) )
3 idd 22 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( ( cf `  x
)  =  x  -> 
( cf `  x
)  =  x ) )
4 pwfi 7364 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 7567 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 8529 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  om  C_  x
)
7 ssdomg 7116 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  Inacc W  ->  om  ~<_  x )
9 sdomdomtr 7203 . . . . . . . . . . . . . . . 16  |-  ( ( ~P y  ~<  om  /\  om  ~<_  x )  ->  ~P y  ~<  x )
109expcom 425 . . . . . . . . . . . . . . 15  |-  ( om  ~<_  x  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x ) )
118, 10syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  Inacc W  ->  ( ~P y  ~<  om  ->  ~P y  ~<  x )
)
125, 11syl5bi 209 . . . . . . . . . . . . 13  |-  ( x  e.  Inacc W  ->  ( ~P y  e.  Fin  ->  ~P y  ~<  x
) )
134, 12syl5bi 209 . . . . . . . . . . . 12  |-  ( x  e.  Inacc W  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1413ad3antlr 712 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ~P y  ~<  x )
)
1514a1dd 44 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x )
) )
16 vex 2923 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 simplll 735 . . . . . . . . . . . . . . 15  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> GCH  =  _V )
1816, 17syl5eleqr 2495 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
19 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
20 gchinf 8492 . . . . . . . . . . . . . 14  |-  ( ( y  e. GCH  /\  -.  y  e.  Fin )  ->  om  ~<_  y )
2118, 19, 20syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  om 
~<_  y )
22 vex 2923 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2495 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 8509 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 8527 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  Inacc W  ->  ( card `  x )  =  x )
27 iscard 7822 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. z  e.  x  z  ~<  x ) )
2827simprbi 451 . . . . . . . . . . . . . . . . 17  |-  ( (
card `  x )  =  x  ->  A. z  e.  x  z  ~<  x )
2926, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  Inacc W  ->  A. z  e.  x  z  ~<  x )
3029ad2antlr 708 . . . . . . . . . . . . . . 15  |-  ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  ->  A. z  e.  x  z  ~<  x )
3130r19.21bi 2768 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7205 . . . . . . . . . . . . . . 15  |-  ( ( ~P y  ~<_  z  /\  z  ~<  x )  ->  ~P y  ~<  x )
3332expcom 425 . . . . . . . . . . . . . 14  |-  ( z 
~<  x  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3431, 33syl 16 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( ~P y  ~<_  z  ->  ~P y  ~<  x )
)
3534adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( ~P y  ~<_  z  ->  ~P y  ~<  x ) )
3625, 35sylbid 207 . . . . . . . . . . 11  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  ->  ~P y  ~<  x
) )
3736expr 599 . . . . . . . . . 10  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  ( -.  y  e.  Fin  ->  ( y  ~<  z  ->  ~P y  ~<  x
) ) )
3815, 37pm2.61d 152 . . . . . . . . 9  |-  ( ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  /\  z  e.  x )  ->  (
y  ~<  z  ->  ~P y  ~<  x ) )
3938rexlimdva 2794 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  Inacc W )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2748 . . . . . . 7  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( A. y  e.  x  E. z  e.  x  y  ~<  z  ->  A. y  e.  x  ~P y  ~<  x ) )
412, 3, 403anim123d 1261 . . . . . 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 ) ) )
42 elwina 8521 . . . . . 6  |-  ( x  e.  Inacc W  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 8522 . . . . . 6  |-  ( x  e.  Inacc 
<->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) )
4441, 42, 433imtr4g 262 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  -> 
( x  e.  Inacc W  ->  x  e.  Inacc ) )
451, 44mpd 15 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  Inacc W )  ->  x  e.  Inacc )
4645ex 424 . . 3  |-  (GCH  =  _V  ->  ( x  e. 
Inacc W  ->  x  e. 
Inacc ) )
47 inawina 8525 . . 3  |-  ( x  e.  Inacc  ->  x  e.  Inacc W )
4846, 47impbid1 195 . 2  |-  (GCH  =  _V  ->  ( x  e. 
Inacc W  <->  x  e.  Inacc ) )
4948eqrdv 2406 1  |-  (GCH  =  _V  ->  Inacc W  =  Inacc )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   _Vcvv 2920    C_ wss 3284   (/)c0 3592   ~Pcpw 3763   class class class wbr 4176   Oncon0 4545   omcom 4808   ` cfv 5417    ~<_ cdom 7070    ~< csdm 7071   Fincfn 7072   cardccrd 7782   cfccf 7784  GCHcgch 8455   Inacc Wcwina 8517   Inacccina 8518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-seqom 6668  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-oexp 6693  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-wdom 7487  df-cnf 7577  df-card 7786  df-cf 7788  df-cda 8008  df-fin4 8127  df-gch 8456  df-wina 8519  df-ina 8520
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