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Theorem gchina 8988
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 459 . . . . 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 7730 . . . . . . . . . . . . 13  |-  ( y  e.  Fin  <->  ~P y  e.  Fin )
5 isfinite 7983 . . . . . . . . . . . . . 14  |-  ( ~P y  e.  Fin  <->  ~P y  ~<  om )
6 winainf 8983 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  om  C_  x
)
7 ssdomg 7480 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  ( om  C_  x  ->  om  ~<_  x ) )
86, 7mpd 15 . . . . . . . . . . . . . . 15  |-  ( x  e.  InaccW  ->  om  ~<_  x )
9 sdomdomtr 7569 . . . . . . . . . . . . . . . 16  |-  ( ( ~P y  ~<  om  /\  om  ~<_  x )  ->  ~P y  ~<  x )
109expcom 433 . . . . . . . . . . . . . . 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 728 . . . . . . . . . . 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 3037 . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
y  e. GCH )
19 simprr 755 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  -.  y  e.  Fin )
20 gchinf 8946 . . . . . . . . . . . . . 14  |-  ( ( y  e. GCH  /\  -.  y  e.  Fin )  ->  om  ~<_  y )
2118, 19, 20syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  ->  om 
~<_  y )
22 vex 3037 . . . . . . . . . . . . . 14  |-  z  e. 
_V
2322, 17syl5eleqr 2477 . . . . . . . . . . . . 13  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
z  e. GCH )
24 gchpwdom 8959 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  y  /\  y  e. GCH  /\  z  e. GCH )  ->  ( y  ~<  z  <->  ~P y  ~<_  z ) )
2521, 18, 23, 24syl3anc 1226 . . . . . . . . . . . 12  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  ( z  e.  x  /\  -.  y  e.  Fin ) )  -> 
( y  ~<  z  <->  ~P y  ~<_  z ) )
26 winacard 8981 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  InaccW  ->  ( card `  x )  =  x )
27 iscard 8269 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. z  e.  x  z  ~<  x ) )
2827simprbi 462 . . . . . . . . . . . . . . . . 17  |-  ( (
card `  x )  =  x  ->  A. z  e.  x  z  ~<  x )
2926, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  InaccW  ->  A. z  e.  x  z  ~<  x )
3029ad2antlr 724 . . . . . . . . . . . . . . 15  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  A. z  e.  x  z  ~<  x )
3130r19.21bi 2751 . . . . . . . . . . . . . 14  |-  ( ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  /\  z  e.  x )  ->  z  ~<  x )
32 domsdomtr 7571 . . . . . . . . . . . . . . 15  |-  ( ( ~P y  ~<_  z  /\  z  ~<  x )  ->  ~P y  ~<  x )
3332expcom 433 . . . . . . . . . . . . . 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 714 . . . . . . . . . . . 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 613 . . . . . . . . . 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 2874 . . . . . . . 8  |-  ( ( (GCH  =  _V  /\  x  e.  InaccW )  /\  y  e.  x
)  ->  ( E. z  e.  x  y  ~<  z  ->  ~P y  ~<  x ) )
4039ralimdva 2790 . . . . . . 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 1304 . . . . . 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 8975 . . . . . 6  |-  ( x  e.  InaccW  <->  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z
) )
43 elina 8976 . . . . . 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 432 . . 3  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  ->  x  e. 
Inacc ) )
47 inawina 8979 . . 3  |-  ( x  e.  Inacc  ->  x  e.  InaccW )
4846, 47impbid1 203 . 2  |-  (GCH  =  _V  ->  ( x  e. 
InaccW  <->  x  e.  Inacc ) )
4948eqrdv 2379 1  |-  (GCH  =  _V  ->  InaccW  =  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   _Vcvv 3034    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   class class class wbr 4367   Oncon0 4792   ` cfv 5496   omcom 6599    ~<_ cdom 7433    ~< csdm 7434   Fincfn 7435   cardccrd 8229   cfccf 8231  GCHcgch 8909   InaccWcwina 8971   Inacccina 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-seqom 7031  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-oexp 7054  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-oi 7850  df-har 7899  df-wdom 7900  df-cnf 7992  df-card 8233  df-cf 8235  df-cda 8461  df-fin4 8580  df-gch 8910  df-wina 8973  df-ina 8974
This theorem is referenced by: (None)
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