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Theorem gchhar 8173
Description: A "local" form of gchac 8175. If  A and  ~P A are GCH-sets, then the Hartogs number of  A is  ~P A (so  ~P A and a fortiori 
A are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchhar  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~~  ~P A
)

Proof of Theorem gchhar
StepHypRef Expression
1 harcl 7159 . . . 4  |-  (har `  A )  e.  On
2 simp3 962 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e. GCH )
3 cdadom3 7698 . . . 4  |-  ( ( (har `  A )  e.  On  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
41, 2, 3sylancr 647 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
5 domnsym 6872 . . . . . . . . 9  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
653ad2ant1 981 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~<  om )
7 isfinite 7237 . . . . . . . 8  |-  ( A  e.  Fin  <->  A  ~<  om )
86, 7sylnibr 298 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  e.  Fin )
9 pwfi 7035 . . . . . . 7  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylnib 297 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  ~P A  e.  Fin )
11 cdadom3 7698 . . . . . . 7  |-  ( ( ~P A  e. GCH  /\  (har `  A )  e.  On )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A )
) )
122, 1, 11sylancl 646 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A ) ) )
13 ovex 5735 . . . . . . . 8  |-  ( ~P A  +c  (har `  A ) )  e. 
_V
1413canth2 6899 . . . . . . 7  |-  ( ~P A  +c  (har `  A ) )  ~<  ~P ( ~P A  +c  (har `  A ) )
15 pwcdaen 7695 . . . . . . . . 9  |-  ( ( ~P A  e. GCH  /\  (har `  A )  e.  On )  ->  ~P ( ~P A  +c  (har `  A ) )  ~~  ( ~P ~P A  X.  ~P (har `  A )
) )
162, 1, 15sylancl 646 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  (har `  A ) )  ~~  ( ~P ~P A  X.  ~P (har `  A )
) )
17 pwexg 4088 . . . . . . . . . . 11  |-  ( ~P A  e. GCH  ->  ~P ~P A  e.  _V )
182, 17syl 17 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ~P A  e.  _V )
19 simp2 961 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  e. GCH )
20 harwdom 7188 . . . . . . . . . . 11  |-  ( A  e. GCH  ->  (har `  A
)  ~<_*  ~P ( A  X.  A ) )
21 wdompwdom 7176 . . . . . . . . . . 11  |-  ( (har
`  A )  ~<_*  ~P ( A  X.  A )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A ) )
2219, 20, 213syl 20 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A
) )
23 xpdom2g 6843 . . . . . . . . . 10  |-  ( ( ~P ~P A  e. 
_V  /\  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A
) )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ( ~P ~P A  X.  ~P ~P ( A  X.  A ) ) )
2418, 22, 23syl2anc 645 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ( ~P ~P A  X.  ~P ~P ( A  X.  A ) ) )
25 xpexg 4707 . . . . . . . . . . . . . 14  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ( A  X.  A )  e.  _V )
2619, 19, 25syl2anc 645 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  e.  _V )
27 pwexg 4088 . . . . . . . . . . . . 13  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
2826, 27syl 17 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  X.  A )  e. 
_V )
29 pwcdaen 7695 . . . . . . . . . . . 12  |-  ( ( ~P A  e. GCH  /\  ~P ( A  X.  A
)  e.  _V )  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) )  ~~  ( ~P ~P A  X.  ~P ~P ( A  X.  A ) ) )
302, 28, 29syl2anc 645 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P ~P A  X.  ~P ~P ( A  X.  A ) ) )
31 ensym 6796 . . . . . . . . . . 11  |-  ( ~P ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ( ~P A  +c  ~P ( A  X.  A
) ) )
3230, 31syl 17 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ( ~P A  +c  ~P ( A  X.  A
) ) )
33 enrefg 6779 . . . . . . . . . . . . . 14  |-  ( ~P A  e. GCH  ->  ~P A  ~~  ~P A )
342, 33syl 17 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ~P A )
35 gchxpidm 8171 . . . . . . . . . . . . . . 15  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
3619, 8, 35syl2anc 645 . . . . . . . . . . . . . 14  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  ~~  A
)
37 pwen 6919 . . . . . . . . . . . . . 14  |-  ( ( A  X.  A ) 
~~  A  ->  ~P ( A  X.  A
)  ~~  ~P A
)
3836, 37syl 17 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  X.  A )  ~~  ~P A )
39 cdaen 7683 . . . . . . . . . . . . 13  |-  ( ( ~P A  ~~  ~P A  /\  ~P ( A  X.  A )  ~~  ~P A )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
4034, 38, 39syl2anc 645 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
41 gchcdaidm 8170 . . . . . . . . . . . . 13  |-  ( ( ~P A  e. GCH  /\  -.  ~P A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
422, 10, 41syl2anc 645 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )
43 entr 6798 . . . . . . . . . . . 12  |-  ( ( ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ( ~P A  +c  ~P A
)  /\  ( ~P A  +c  ~P A ) 
~~  ~P A )  -> 
( ~P A  +c  ~P ( A  X.  A
) )  ~~  ~P A )
4440, 42, 43syl2anc 645 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P A )
45 pwen 6919 . . . . . . . . . . 11  |-  ( ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ~P A  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
4644, 45syl 17 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
47 entr 6798 . . . . . . . . . 10  |-  ( ( ( ~P ~P A  X.  ~P ~P ( A  X.  A ) ) 
~~  ~P ( ~P A  +c  ~P ( A  X.  A ) )  /\  ~P ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ~P ~P A )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )
4832, 46, 47syl2anc 645 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )
49 domentr 6805 . . . . . . . . 9  |-  ( ( ( ~P ~P A  X.  ~P (har `  A
) )  ~<_  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  /\  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ~P ~P A )
5024, 48, 49syl2anc 645 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ~P ~P A )
51 endomtr 6804 . . . . . . . 8  |-  ( ( ~P ( ~P A  +c  (har `  A )
)  ~~  ( ~P ~P A  X.  ~P (har `  A ) )  /\  ( ~P ~P A  X.  ~P (har `  A )
)  ~<_  ~P ~P A )  ->  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )
5216, 50, 51syl2anc 645 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )
53 sdomdomtr 6879 . . . . . . 7  |-  ( ( ( ~P A  +c  (har `  A ) ) 
~<  ~P ( ~P A  +c  (har `  A )
)  /\  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )  -> 
( ~P A  +c  (har `  A ) ) 
~<  ~P ~P A )
5414, 52, 53sylancr 647 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<  ~P ~P A )
55 gchen1 8127 . . . . . 6  |-  ( ( ( ~P A  e. GCH  /\  -.  ~P A  e. 
Fin )  /\  ( ~P A  ~<_  ( ~P A  +c  (har `  A
) )  /\  ( ~P A  +c  (har `  A ) )  ~<  ~P ~P A ) )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
562, 10, 12, 54, 55syl22anc 1188 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
57 cdacomen 7691 . . . . 5  |-  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A )
58 entr 6798 . . . . 5  |-  ( ( ~P A  ~~  ( ~P A  +c  (har `  A ) )  /\  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A ) )  ->  ~P A  ~~  ( (har `  A )  +c  ~P A ) )
5956, 57, 58sylancl 646 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( (har `  A
)  +c  ~P A
) )
60 ensym 6796 . . . 4  |-  ( ~P A  ~~  ( (har
`  A )  +c 
~P A )  -> 
( (har `  A
)  +c  ~P A
)  ~~  ~P A
)
6159, 60syl 17 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( (har `  A )  +c  ~P A )  ~~  ~P A )
62 domentr 6805 . . 3  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  ~P A )  /\  ( (har `  A )  +c  ~P A )  ~~  ~P A )  ->  (har `  A )  ~<_  ~P A
)
634, 61, 62syl2anc 645 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ~P A )
64 gchcdaidm 8170 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
6519, 8, 64syl2anc 645 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  A )  ~~  A
)
66 pwen 6919 . . . . 5  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
6765, 66syl 17 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ~P A )
68 cdadom3 7698 . . . . . . . 8  |-  ( ( A  e. GCH  /\  (har `  A )  e.  On )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
6919, 1, 68sylancl 646 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
70 harndom 7162 . . . . . . . 8  |-  -.  (har `  A )  ~<_  A
71 cdadom3 7698 . . . . . . . . . . 11  |-  ( ( (har `  A )  e.  On  /\  A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
721, 19, 71sylancr 647 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
73 cdacomen 7691 . . . . . . . . . 10  |-  ( (har
`  A )  +c  A )  ~~  ( A  +c  (har `  A
) )
74 domentr 6805 . . . . . . . . . 10  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  A )  /\  (
(har `  A )  +c  A )  ~~  ( A  +c  (har `  A
) ) )  -> 
(har `  A )  ~<_  ( A  +c  (har `  A ) ) )
7572, 73, 74sylancl 646 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) )
76 domen2 6889 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  (har `  A ) )  ->  ( (har `  A )  ~<_  A  <->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) ) )
7775, 76syl5ibrcom 215 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  ~~  ( A  +c  (har `  A ) )  -> 
(har `  A )  ~<_  A ) )
7870, 77mtoi 171 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~~  ( A  +c  (har `  A ) ) )
79 brsdom 6770 . . . . . . 7  |-  ( A 
~<  ( A  +c  (har `  A ) )  <->  ( A  ~<_  ( A  +c  (har `  A ) )  /\  -.  A  ~~  ( A  +c  (har `  A
) ) ) )
8069, 78, 79sylanbrc 648 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<  ( A  +c  (har `  A ) ) )
81 canth2g 6900 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
82 sdomdom 6775 . . . . . . . . . 10  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
8319, 81, 823syl 20 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ~P A
)
84 cdadom1 7696 . . . . . . . . 9  |-  ( A  ~<_  ~P A  ->  ( A  +c  (har `  A
) )  ~<_  ( ~P A  +c  (har `  A ) ) )
8583, 84syl 17 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  (har `  A )
) )
86 cdadom2 7697 . . . . . . . . 9  |-  ( (har
`  A )  ~<_  ~P A  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
8763, 86syl 17 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
88 domtr 6799 . . . . . . . 8  |-  ( ( ( A  +c  (har `  A ) )  ~<_  ( ~P A  +c  (har `  A ) )  /\  ( ~P A  +c  (har `  A ) )  ~<_  ( ~P A  +c  ~P A ) )  -> 
( A  +c  (har `  A ) )  ~<_  ( ~P A  +c  ~P A ) )
8985, 87, 88syl2anc 645 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  ~P A ) )
90 domentr 6805 . . . . . . 7  |-  ( ( ( A  +c  (har `  A ) )  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A
)  ~~  ~P A
)  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
9189, 42, 90syl2anc 645 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
92 gchen2 8128 . . . . . 6  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  ( A  +c  (har `  A ) )  /\  ( A  +c  (har `  A ) )  ~<_  ~P A ) )  -> 
( A  +c  (har `  A ) )  ~~  ~P A )
9319, 8, 80, 91, 92syl22anc 1188 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~~  ~P A
)
94 ensym 6796 . . . . 5  |-  ( ( A  +c  (har `  A ) )  ~~  ~P A  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
9593, 94syl 17 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
96 entr 6798 . . . 4  |-  ( ( ~P ( A  +c  A )  ~~  ~P A  /\  ~P A  ~~  ( A  +c  (har `  A ) ) )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
9767, 95, 96syl2anc 645 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
98 endom 6774 . . 3  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  (har `  A )
)  ->  ~P ( A  +c  A )  ~<_  ( A  +c  (har `  A ) ) )
99 pwcdadom 7726 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  (har `  A ) )  ->  ~P A  ~<_  (har
`  A ) )
10097, 98, 993syl 20 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  (har `  A ) )
101 sbth 6866 . 2  |-  ( ( (har `  A )  ~<_  ~P A  /\  ~P A  ~<_  (har `  A ) )  ->  (har `  A
)  ~~  ~P A
)
10263, 100, 101syl2anc 645 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~~  ~P A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ w3a 939    e. wcel 1621   _Vcvv 2727   ~Pcpw 3530   class class class wbr 3920   Oncon0 4285   omcom 4547    X. cxp 4578   ` cfv 4592  (class class class)co 5710    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748   Fincfn 6749  harchar 7154    ~<_* cwdom 7155    +c ccda 7677  GCHcgch 8122
This theorem is referenced by:  gchacg  8174
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-seqom 6346  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-oexp 6371  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-wdom 7157  df-cnf 7247  df-card 7456  df-cda 7678  df-fin4 7797  df-gch 8123
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