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Theorem gchhar 8538
Description: A "local" form of gchac 8540. 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 7521 . . . 4  |-  (har `  A )  e.  On
2 simp3 959 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e. GCH )
3 cdadom3 8060 . . . 4  |-  ( ( (har `  A )  e.  On  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
41, 2, 3sylancr 645 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  ~P A ) )
5 domnsym 7225 . . . . . . . . 9  |-  ( om  ~<_  A  ->  -.  A  ~<  om )
653ad2ant1 978 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~<  om )
7 isfinite 7599 . . . . . . . 8  |-  ( A  e.  Fin  <->  A  ~<  om )
86, 7sylnibr 297 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  e.  Fin )
9 pwfi 7394 . . . . . . 7  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylnib 296 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  ~P A  e.  Fin )
11 cdadom3 8060 . . . . . . 7  |-  ( ( ~P A  e. GCH  /\  (har `  A )  e.  On )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A )
) )
122, 1, 11sylancl 644 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  ( ~P A  +c  (har `  A ) ) )
13 ovex 6098 . . . . . . . 8  |-  ( ~P A  +c  (har `  A ) )  e. 
_V
1413canth2 7252 . . . . . . 7  |-  ( ~P A  +c  (har `  A ) )  ~<  ~P ( ~P A  +c  (har `  A ) )
15 pwcdaen 8057 . . . . . . . . 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 644 . . . . . . . 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 4375 . . . . . . . . . . 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 958 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  e. GCH )
20 harwdom 7550 . . . . . . . . . . 11  |-  ( A  e. GCH  ->  (har `  A
)  ~<_*  ~P ( A  X.  A ) )
21 wdompwdom 7538 . . . . . . . . . . 11  |-  ( (har
`  A )  ~<_*  ~P ( A  X.  A )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A ) )
2219, 20, 213syl 19 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P (har `  A )  ~<_  ~P ~P ( A  X.  A
) )
23 xpdom2g 7196 . . . . . . . . . 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 643 . . . . . . . . 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 4981 . . . . . . . . . . . . . 14  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ( A  X.  A )  e.  _V )
2619, 19, 25syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  e.  _V )
27 pwexg 4375 . . . . . . . . . . . . 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 8057 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 7150 . . . . . . . . . 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 7131 . . . . . . . . . . . . . 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 8536 . . . . . . . . . . . . . . 15  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
3519, 8, 34syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  X.  A )  ~~  A
)
36 pwen 7272 . . . . . . . . . . . . . 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 8045 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ( ~P A  +c  ~P A ) )
40 gchcdaidm 8535 . . . . . . . . . . . . 13  |-  ( ( ~P A  e. GCH  /\  -.  ~P A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
412, 10, 40syl2anc 643 . . . . . . . . . . . 12  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )
42 entr 7151 . . . . . . . . . . . 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 643 . . . . . . . . . . 11  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  ~P ( A  X.  A ) ) 
~~  ~P A )
44 pwen 7272 . . . . . . . . . . 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 7151 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P ~P ( A  X.  A
) )  ~~  ~P ~P A )
48 domentr 7158 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P ~P A  X.  ~P (har `  A ) )  ~<_  ~P ~P A )
50 endomtr 7157 . . . . . . . 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 643 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( ~P A  +c  (har `  A ) )  ~<_  ~P ~P A )
52 sdomdomtr 7232 . . . . . . 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 645 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<  ~P ~P A )
54 gchen1 8492 . . . . . 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 1185 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( ~P A  +c  (har `  A ) ) )
56 cdacomen 8053 . . . . 5  |-  ( ~P A  +c  (har `  A ) )  ~~  ( (har `  A )  +c  ~P A )
57 entr 7151 . . . . 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 644 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( (har `  A
)  +c  ~P A
) )
5958ensymd 7150 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( (har `  A )  +c  ~P A )  ~~  ~P A )
60 domentr 7158 . . 3  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  ~P A )  /\  ( (har `  A )  +c  ~P A )  ~~  ~P A )  ->  (har `  A )  ~<_  ~P A
)
614, 59, 60syl2anc 643 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ~P A )
62 gchcdaidm 8535 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
6319, 8, 62syl2anc 643 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  A )  ~~  A
)
64 pwen 7272 . . . . 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 8060 . . . . . . . 8  |-  ( ( A  e. GCH  /\  (har `  A )  e.  On )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
6719, 1, 66sylancl 644 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ( A  +c  (har `  A
) ) )
68 harndom 7524 . . . . . . . 8  |-  -.  (har `  A )  ~<_  A
69 cdadom3 8060 . . . . . . . . . . 11  |-  ( ( (har `  A )  e.  On  /\  A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
701, 19, 69sylancr 645 . . . . . . . . . 10  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( (har `  A )  +c  A
) )
71 cdacomen 8053 . . . . . . . . . 10  |-  ( (har
`  A )  +c  A )  ~~  ( A  +c  (har `  A
) )
72 domentr 7158 . . . . . . . . . 10  |-  ( ( (har `  A )  ~<_  ( (har `  A )  +c  A )  /\  (
(har `  A )  +c  A )  ~~  ( A  +c  (har `  A
) ) )  -> 
(har `  A )  ~<_  ( A  +c  (har `  A ) ) )
7370, 71, 72sylancl 644 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) )
74 domen2 7242 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  (har `  A ) )  ->  ( (har `  A )  ~<_  A  <->  (har `  A
)  ~<_  ( A  +c  (har `  A ) ) ) )
7573, 74syl5ibrcom 214 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  ~~  ( A  +c  (har `  A ) )  -> 
(har `  A )  ~<_  A ) )
7668, 75mtoi 171 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  -.  A  ~~  ( A  +c  (har `  A ) ) )
77 brsdom 7122 . . . . . . 7  |-  ( A 
~<  ( A  +c  (har `  A ) )  <->  ( A  ~<_  ( A  +c  (har `  A ) )  /\  -.  A  ~~  ( A  +c  (har `  A
) ) ) )
7867, 76, 77sylanbrc 646 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<  ( A  +c  (har `  A ) ) )
79 canth2g 7253 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
80 sdomdom 7127 . . . . . . . . . 10  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
8119, 79, 803syl 19 . . . . . . . . 9  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  A  ~<_  ~P A
)
82 cdadom1 8058 . . . . . . . . 9  |-  ( A  ~<_  ~P A  ->  ( A  +c  (har `  A
) )  ~<_  ( ~P A  +c  (har `  A ) ) )
8381, 82syl 16 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  (har `  A )
) )
84 cdadom2 8059 . . . . . . . . 9  |-  ( (har
`  A )  ~<_  ~P A  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
8561, 84syl 16 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( ~P A  +c  (har `  A
) )  ~<_  ( ~P A  +c  ~P A
) )
86 domtr 7152 . . . . . . . 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 ) )
8783, 85, 86syl2anc 643 . . . . . . 7  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ( ~P A  +c  ~P A ) )
88 domentr 7158 . . . . . . 7  |-  ( ( ( A  +c  (har `  A ) )  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A
)  ~~  ~P A
)  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
8987, 41, 88syl2anc 643 . . . . . 6  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~<_  ~P A )
90 gchen2 8493 . . . . . 6  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  ( A  +c  (har `  A ) )  /\  ( A  +c  (har `  A ) )  ~<_  ~P A ) )  -> 
( A  +c  (har `  A ) )  ~~  ~P A )
9119, 8, 78, 89, 90syl22anc 1185 . . . . 5  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ( A  +c  (har `  A )
)  ~~  ~P A
)
9291ensymd 7150 . . . 4  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~~  ( A  +c  (har `  A ) ) )
93 entr 7151 . . . 4  |-  ( ( ~P ( A  +c  A )  ~~  ~P A  /\  ~P A  ~~  ( A  +c  (har `  A ) ) )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
9465, 92, 93syl2anc 643 . . 3  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( A  +c  (har `  A ) ) )
95 endom 7126 . . 3  |-  ( ~P ( A  +c  A
)  ~~  ( A  +c  (har `  A )
)  ->  ~P ( A  +c  A )  ~<_  ( A  +c  (har `  A ) ) )
96 pwcdadom 8088 . . 3  |-  ( ~P ( A  +c  A
)  ~<_  ( A  +c  (har `  A ) )  ->  ~P A  ~<_  (har
`  A ) )
9794, 95, 963syl 19 . 2  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  ~<_  (har `  A ) )
98 sbth 7219 . 2  |-  ( ( (har `  A )  ~<_  ~P A  /\  ~P A  ~<_  (har `  A ) )  ->  (har `  A
)  ~~  ~P A
)
9961, 97, 98syl2anc 643 1  |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A
)  ~~  ~P A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 936    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204   Oncon0 4573   omcom 4837    X. cxp 4868   ` cfv 5446  (class class class)co 6073    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   Fincfn 7101  harchar 7516    ~<_* cwdom 7517    +c ccda 8039  GCHcgch 8487
This theorem is referenced by:  gchacg  8539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-wdom 7519  df-cnf 7609  df-card 7818  df-cda 8040  df-fin4 8159  df-gch 8488
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