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Theorem gchhar 9053
Description: A "local" form of gchac 9055. 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 7983 . . . 4  |-  (har `  A )  e.  On
2 simp3 998 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e. GCH )
3 cdadom3 8564 . . . 4  |-  ( ( (har `  A )  e.  On  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
41, 2, 3sylancr 663 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
5 domnsym 7640 . . . . . . . . 9  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
653ad2ant1 1017 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~<  om )
7 isfinite 8065 . . . . . . . 8  |-  ( A  e.  Fin  <->  A  ~<  om )
86, 7sylnibr 305 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  e.  Fin )
9 pwfi 7811 . . . . . . 7  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylnib 304 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  ~P A  e.  Fin )
11 cdadom3 8564 . . . . . . 7  |-  ( ( ~P A  e. GCH  /\  (har `  A )  e.  On )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A )
) )
122, 1, 11sylancl 662 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A ) ) )
13 ovex 6307 . . . . . . . 8  |-  ( ~P A  +c  (har `  A ) )  e. 
_V
1413canth2 7667 . . . . . . 7  |-  ( ~P A  +c  (har `  A ) )  ~<  ~P ( ~P A  +c  (har `  A ) )
15 pwcdaen 8561 . . . . . . . . 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 662 . . . . . . . 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 4631 . . . . . . . . . . 11  |-  ( ~P A  e. GCH  ->  ~P ~P A  e.  _V )
182, 17syl 16 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ~P A  e.  _V )
19 simp2 997 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  e. GCH )
20 harwdom 8012 . . . . . . . . . . 11  |-  ( A  e. GCH  ->  (har `  A
)  ~<_*  ~P ( A  X.  A ) )
21 wdompwdom 8000 . . . . . . . . . . 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 7610 . . . . . . . . . 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 661 . . . . . . . . 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 6709 . . . . . . . . . . . . . 14  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ( A  X.  A )  e.  _V )
2619, 19, 25syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  e.  _V )
27 pwexg 4631 . . . . . . . . . . . . 13  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
2826, 27syl 16 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  X.  A )  e. 
_V )
29 pwcdaen 8561 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 7563 . . . . . . . . . 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 7544 . . . . . . . . . . . . . 14  |-  ( ~P A  e. GCH  ->  ~P A  ~~  ~P A )
332, 32syl 16 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ~P A )
34 gchxpidm 9043 . . . . . . . . . . . . . . 15  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
3519, 8, 34syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  ~~  A
)
36 pwen 7687 . . . . . . . . . . . . . 14  |-  ( ( A  X.  A ) 
~~  A  ->  ~P ( A  X.  A
)  ~~  ~P A
)
3735, 36syl 16 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  X.  A )  ~~  ~P A )
38 cdaen 8549 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
40 gchcdaidm 9042 . . . . . . . . . . . . 13  |-  ( ( ~P A  e. GCH  /\  -.  ~P A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
412, 10, 40syl2anc 661 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )
42 entr 7564 . . . . . . . . . . . 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 661 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P A )
44 pwen 7687 . . . . . . . . . . 11  |-  ( ( ~P A  +c  ~P ( A  X.  A
) )  ~~  ~P A  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
4543, 44syl 16 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P ~P A )
46 entr 7564 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )
48 domentr 7571 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ~P ~P A )
50 endomtr 7570 . . . . . . . 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 661 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )
52 sdomdomtr 7647 . . . . . . 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 663 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<  ~P ~P A )
54 gchen1 8999 . . . . . 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 1229 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
56 cdacomen 8557 . . . . 5  |-  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A )
57 entr 7564 . . . . 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 662 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( (har `  A
)  +c  ~P A
) )
5958ensymd 7563 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( (har `  A )  +c  ~P A )  ~~  ~P A )
60 domentr 7571 . . 3  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  ~P A )  /\  ( (har `  A )  +c  ~P A )  ~~  ~P A )  ->  (har `  A )  ~<_  ~P A
)
614, 59, 60syl2anc 661 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ~P A )
62 gchcdaidm 9042 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
6319, 8, 62syl2anc 661 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  A )  ~~  A
)
64 pwen 7687 . . . . 5  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
6563, 64syl 16 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ~P A )
66 cdadom3 8564 . . . . . . . 8  |-  ( ( A  e. GCH  /\  (har `  A )  e.  On )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
6719, 1, 66sylancl 662 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
68 harndom 7986 . . . . . . . 8  |-  -.  (har `  A )  ~<_  A
69 cdadom3 8564 . . . . . . . . . . 11  |-  ( ( (har `  A )  e.  On  /\  A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
701, 19, 69sylancr 663 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
71 cdacomen 8557 . . . . . . . . . 10  |-  ( (har
`  A )  +c  A )  ~~  ( A  +c  (har `  A
) )
72 domentr 7571 . . . . . . . . . 10  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  A )  /\  (
(har `  A )  +c  A )  ~~  ( A  +c  (har `  A
) ) )  -> 
(har `  A )  ~<_  ( A  +c  (har `  A ) ) )
7370, 71, 72sylancl 662 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) )
74 domen2 7657 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  (har `  A ) )  ->  ( (har `  A )  ~<_  A  <->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) ) )
7573, 74syl5ibrcom 222 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  ~~  ( A  +c  (har `  A ) )  -> 
(har `  A )  ~<_  A ) )
7668, 75mtoi 178 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~~  ( A  +c  (har `  A ) ) )
77 brsdom 7535 . . . . . . 7  |-  ( A 
~<  ( A  +c  (har `  A ) )  <->  ( A  ~<_  ( A  +c  (har `  A ) )  /\  -.  A  ~~  ( A  +c  (har `  A
) ) ) )
7867, 76, 77sylanbrc 664 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<  ( A  +c  (har `  A ) ) )
79 canth2g 7668 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
80 sdomdom 7540 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
81 cdadom1 8562 . . . . . . . . 9  |-  ( A  ~<_  ~P A  ->  ( A  +c  (har `  A
) )  ~<_  ( ~P A  +c  (har `  A ) ) )
8219, 79, 80, 814syl 21 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  (har `  A )
) )
83 cdadom2 8563 . . . . . . . . 9  |-  ( (har
`  A )  ~<_  ~P A  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
8461, 83syl 16 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
85 domtr 7565 . . . . . . . 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 661 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  ~P A ) )
87 domentr 7571 . . . . . . 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 661 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
89 gchen2 9000 . . . . . 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 1229 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~~  ~P A
)
9190ensymd 7563 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
92 entr 7564 . . . 4  |-  ( ( ~P ( A  +c  A )  ~~  ~P A  /\  ~P A  ~~  ( A  +c  (har `  A ) ) )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
9365, 91, 92syl2anc 661 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
94 endom 7539 . . 3  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  (har `  A )
)  ->  ~P ( A  +c  A )  ~<_  ( A  +c  (har `  A ) ) )
95 pwcdadom 8592 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  (har `  A ) )  ->  ~P A  ~<_  (har
`  A ) )
9693, 94, 953syl 20 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  (har `  A ) )
97 sbth 7634 . 2  |-  ( ( (har `  A )  ~<_  ~P A  /\  ~P A  ~<_  (har `  A ) )  ->  (har `  A
)  ~~  ~P A
)
9861, 96, 97syl2anc 661 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 973    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447   Oncon0 4878    X. cxp 4997   ` cfv 5586  (class class class)co 6282   omcom 6678    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512   Fincfn 7513  harchar 7978    ~<_* cwdom 7979    +c ccda 8543  GCHcgch 8994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-oexp 7133  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-har 7980  df-wdom 7981  df-cnf 8075  df-card 8316  df-cda 8544  df-fin4 8663  df-gch 8995
This theorem is referenced by:  gchacg  9054
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