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Theorem gchhar 9106
Description: A "local" form of gchac 9108. 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 8080 . . . 4  |-  (har `  A )  e.  On
2 simp3 1008 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e. GCH )
3 cdadom3 8620 . . . 4  |-  ( ( (har `  A )  e.  On  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
41, 2, 3sylancr 668 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
5 domnsym 7702 . . . . . . . . 9  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
653ad2ant1 1027 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~<  om )
7 isfinite 8161 . . . . . . . 8  |-  ( A  e.  Fin  <->  A  ~<  om )
86, 7sylnibr 307 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  e.  Fin )
9 pwfi 7873 . . . . . . 7  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylnib 306 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  ~P A  e.  Fin )
11 cdadom3 8620 . . . . . . 7  |-  ( ( ~P A  e. GCH  /\  (har `  A )  e.  On )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A )
) )
122, 1, 11sylancl 667 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A ) ) )
13 ovex 6331 . . . . . . . 8  |-  ( ~P A  +c  (har `  A ) )  e. 
_V
1413canth2 7729 . . . . . . 7  |-  ( ~P A  +c  (har `  A ) )  ~<  ~P ( ~P A  +c  (har `  A ) )
15 pwcdaen 8617 . . . . . . . . 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 667 . . . . . . . 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 4606 . . . . . . . . . . 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 1007 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  e. GCH )
20 harwdom 8109 . . . . . . . . . . 11  |-  ( A  e. GCH  ->  (har `  A
)  ~<_*  ~P ( A  X.  A ) )
21 wdompwdom 8097 . . . . . . . . . . 11  |-  ( (har
`  A )  ~<_*  ~P ( A  X.  A )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A ) )
2219, 20, 213syl 18 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A
) )
23 xpdom2g 7672 . . . . . . . . . 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 666 . . . . . . . . 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 6605 . . . . . . . . . . . . . 14  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ( A  X.  A )  e.  _V )
2619, 19, 25syl2anc 666 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  e.  _V )
27 pwexg 4606 . . . . . . . . . . . . 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 8617 . . . . . . . . . . . 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 666 . . . . . . . . . . 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 ) ) )
3130ensymd 7625 . . . . . . . . . 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
) ) )
32 enrefg 7606 . . . . . . . . . . . . . 14  |-  ( ~P A  e. GCH  ->  ~P A  ~~  ~P A )
332, 32syl 17 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ~P A )
34 gchxpidm 9096 . . . . . . . . . . . . . . 15  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
3519, 8, 34syl2anc 666 . . . . . . . . . . . . . 14  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  ~~  A
)
36 pwen 7749 . . . . . . . . . . . . . 14  |-  ( ( A  X.  A ) 
~~  A  ->  ~P ( A  X.  A
)  ~~  ~P A
)
3735, 36syl 17 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  X.  A )  ~~  ~P A )
38 cdaen 8605 . . . . . . . . . . . . 13  |-  ( ( ~P A  ~~  ~P A  /\  ~P ( A  X.  A )  ~~  ~P A )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
3933, 37, 38syl2anc 666 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
40 gchcdaidm 9095 . . . . . . . . . . . . 13  |-  ( ( ~P A  e. GCH  /\  -.  ~P A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
412, 10, 40syl2anc 666 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )
42 entr 7626 . . . . . . . . . . . 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 )
4339, 41, 42syl2anc 666 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P A )
44 pwen 7749 . . . . . . . . . . 11  |-  ( ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ~P A  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
4543, 44syl 17 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
46 entr 7626 . . . . . . . . . 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 )
4731, 45, 46syl2anc 666 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )
48 domentr 7633 . . . . . . . . 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 )
4924, 47, 48syl2anc 666 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ~P ~P A )
50 endomtr 7632 . . . . . . . 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 )
5116, 49, 50syl2anc 666 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )
52 sdomdomtr 7709 . . . . . . 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 )
5314, 51, 52sylancr 668 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<  ~P ~P A )
54 gchen1 9052 . . . . . 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 ) ) )
552, 10, 12, 53, 54syl22anc 1266 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
56 cdacomen 8613 . . . . 5  |-  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A )
57 entr 7626 . . . . 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 ) )
5855, 56, 57sylancl 667 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( (har `  A
)  +c  ~P A
) )
5958ensymd 7625 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( (har `  A )  +c  ~P A )  ~~  ~P A )
60 domentr 7633 . . 3  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  ~P A )  /\  ( (har `  A )  +c  ~P A )  ~~  ~P A )  ->  (har `  A )  ~<_  ~P A
)
614, 59, 60syl2anc 666 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ~P A )
62 gchcdaidm 9095 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
6319, 8, 62syl2anc 666 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  A )  ~~  A
)
64 pwen 7749 . . . . 5  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
6563, 64syl 17 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ~P A )
66 cdadom3 8620 . . . . . . . 8  |-  ( ( A  e. GCH  /\  (har `  A )  e.  On )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
6719, 1, 66sylancl 667 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
68 harndom 8083 . . . . . . . 8  |-  -.  (har `  A )  ~<_  A
69 cdadom3 8620 . . . . . . . . . . 11  |-  ( ( (har `  A )  e.  On  /\  A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
701, 19, 69sylancr 668 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
71 cdacomen 8613 . . . . . . . . . 10  |-  ( (har
`  A )  +c  A )  ~~  ( A  +c  (har `  A
) )
72 domentr 7633 . . . . . . . . . 10  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  A )  /\  (
(har `  A )  +c  A )  ~~  ( A  +c  (har `  A
) ) )  -> 
(har `  A )  ~<_  ( A  +c  (har `  A ) ) )
7370, 71, 72sylancl 667 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) )
74 domen2 7719 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  (har `  A ) )  ->  ( (har `  A )  ~<_  A  <->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) ) )
7573, 74syl5ibrcom 226 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  ~~  ( A  +c  (har `  A ) )  -> 
(har `  A )  ~<_  A ) )
7668, 75mtoi 182 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~~  ( A  +c  (har `  A ) ) )
77 brsdom 7597 . . . . . . 7  |-  ( A 
~<  ( A  +c  (har `  A ) )  <->  ( A  ~<_  ( A  +c  (har `  A ) )  /\  -.  A  ~~  ( A  +c  (har `  A
) ) ) )
7867, 76, 77sylanbrc 669 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<  ( A  +c  (har `  A ) ) )
79 canth2g 7730 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
80 sdomdom 7602 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
81 cdadom1 8618 . . . . . . . . 9  |-  ( A  ~<_  ~P A  ->  ( A  +c  (har `  A
) )  ~<_  ( ~P A  +c  (har `  A ) ) )
8219, 79, 80, 814syl 19 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  (har `  A )
) )
83 cdadom2 8619 . . . . . . . . 9  |-  ( (har
`  A )  ~<_  ~P A  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
8461, 83syl 17 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
85 domtr 7627 . . . . . . . 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 ) )
8682, 84, 85syl2anc 666 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  ~P A ) )
87 domentr 7633 . . . . . . 7  |-  ( ( ( A  +c  (har `  A ) )  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A
)  ~~  ~P A
)  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
8886, 41, 87syl2anc 666 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
89 gchen2 9053 . . . . . 6  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  ( A  +c  (har `  A ) )  /\  ( A  +c  (har `  A ) )  ~<_  ~P A ) )  -> 
( A  +c  (har `  A ) )  ~~  ~P A )
9019, 8, 78, 88, 89syl22anc 1266 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~~  ~P A
)
9190ensymd 7625 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
92 entr 7626 . . . 4  |-  ( ( ~P ( A  +c  A )  ~~  ~P A  /\  ~P A  ~~  ( A  +c  (har `  A ) ) )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
9365, 91, 92syl2anc 666 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
94 endom 7601 . . 3  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  (har `  A )
)  ->  ~P ( A  +c  A )  ~<_  ( A  +c  (har `  A ) ) )
95 pwcdadom 8648 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  (har `  A ) )  ->  ~P A  ~<_  (har
`  A ) )
9693, 94, 953syl 18 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  (har `  A ) )
97 sbth 7696 . 2  |-  ( ( (har `  A )  ~<_  ~P A  /\  ~P A  ~<_  (har `  A ) )  ->  (har `  A
)  ~~  ~P A
)
9861, 96, 97syl2anc 666 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~~  ~P A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 983    e. wcel 1869   _Vcvv 3082   ~Pcpw 3980   class class class wbr 4421    X. cxp 4849   Oncon0 5440   ` cfv 5599  (class class class)co 6303   omcom 6704    ~~ cen 7572    ~<_ cdom 7573    ~< csdm 7574   Fincfn 7575  harchar 8075    ~<_* cwdom 8076    +c ccda 8599  GCHcgch 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-seqom 7171  df-1o 7188  df-2o 7189  df-oadd 7192  df-omul 7193  df-oexp 7194  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-oi 8029  df-har 8077  df-wdom 8078  df-cnf 8170  df-card 8376  df-cda 8600  df-fin4 8719  df-gch 9048
This theorem is referenced by:  gchacg  9107
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