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Theorem gchhar 8851
Description: A "local" form of gchac 8853. 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 7781 . . . 4  |-  (har `  A )  e.  On
2 simp3 990 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e. GCH )
3 cdadom3 8362 . . . 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 7442 . . . . . . . . 9  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
653ad2ant1 1009 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~<  om )
7 isfinite 7863 . . . . . . . 8  |-  ( A  e.  Fin  <->  A  ~<  om )
86, 7sylnibr 305 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  e.  Fin )
9 pwfi 7611 . . . . . . 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 8362 . . . . . . 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 6121 . . . . . . . 8  |-  ( ~P A  +c  (har `  A ) )  e. 
_V
1413canth2 7469 . . . . . . 7  |-  ( ~P A  +c  (har `  A ) )  ~<  ~P ( ~P A  +c  (har `  A ) )
15 pwcdaen 8359 . . . . . . . . 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 4481 . . . . . . . . . . 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 989 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  e. GCH )
20 harwdom 7810 . . . . . . . . . . 11  |-  ( A  e. GCH  ->  (har `  A
)  ~<_*  ~P ( A  X.  A ) )
21 wdompwdom 7798 . . . . . . . . . . 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 7412 . . . . . . . . . 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 6512 . . . . . . . . . . . . . 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 4481 . . . . . . . . . . . . 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 8359 . . . . . . . . . . . 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 7365 . . . . . . . . . 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 7346 . . . . . . . . . . . . . 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 8841 . . . . . . . . . . . . . . 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 7489 . . . . . . . . . . . . . 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 8347 . . . . . . . . . . . . 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 8840 . . . . . . . . . . . . 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 7366 . . . . . . . . . . . 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 7489 . . . . . . . . . . 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 7366 . . . . . . . . . 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 7373 . . . . . . . . 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 7372 . . . . . . . 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 7449 . . . . . . 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 8797 . . . . . 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 1219 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
56 cdacomen 8355 . . . . 5  |-  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A )
57 entr 7366 . . . . 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 7365 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( (har `  A )  +c  ~P A )  ~~  ~P A )
60 domentr 7373 . . 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 8840 . . . . . 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 7489 . . . . 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 8362 . . . . . . . 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 7784 . . . . . . . 8  |-  -.  (har `  A )  ~<_  A
69 cdadom3 8362 . . . . . . . . . . 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 8355 . . . . . . . . . 10  |-  ( (har
`  A )  +c  A )  ~~  ( A  +c  (har `  A
) )
72 domentr 7373 . . . . . . . . . 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 7459 . . . . . . . . 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 7337 . . . . . . 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 7470 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
80 sdomdom 7342 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
81 cdadom1 8360 . . . . . . . . 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 8361 . . . . . . . . 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 7367 . . . . . . . 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 7373 . . . . . . 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 8798 . . . . . 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 1219 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~~  ~P A
)
9190ensymd 7365 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
92 entr 7366 . . . 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 7341 . . 3  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  (har `  A )
)  ->  ~P ( A  +c  A )  ~<_  ( A  +c  (har `  A ) ) )
95 pwcdadom 8390 . . 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 7436 . 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 965    e. wcel 1756   _Vcvv 2977   ~Pcpw 3865   class class class wbr 4297   Oncon0 4724    X. cxp 4843   ` cfv 5423  (class class class)co 6096   omcom 6481    ~~ cen 7312    ~<_ cdom 7313    ~< csdm 7314   Fincfn 7315  harchar 7776    ~<_* cwdom 7777    +c ccda 8341  GCHcgch 8792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-oexp 6931  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-har 7778  df-wdom 7779  df-cnf 7873  df-card 8114  df-cda 8342  df-fin4 8461  df-gch 8793
This theorem is referenced by:  gchacg  8852
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