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Theorem dominf 8835
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8825. See dominfac 8958 for a version proved from ax-ac 8849. The axiom of Regularity is used for this proof, via inf3lem6 8060, and its use is necessary: otherwise the set  A  =  { A } or  A  =  { (/)
,  A } (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
Hypothesis
Ref Expression
dominf.1  |-  A  e. 
_V
Assertion
Ref Expression
dominf  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )

Proof of Theorem dominf
Dummy variables  x  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dominf.1 . 2  |-  A  e. 
_V
2 neeq1 2748 . . . 4  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 unieq 4258 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
53, 4sseq12d 3538 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. x  <->  A 
C_  U. A ) )
62, 5anbi12d 710 . . 3  |-  ( x  =  A  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  <->  ( A  =/=  (/)  /\  A  C_  U. A ) ) )
7 breq2 4456 . . 3  |-  ( x  =  A  ->  ( om 
~<_  x  <->  om  ~<_  A ) )
86, 7imbi12d 320 . 2  |-  ( x  =  A  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  ~<_  x )  <-> 
( ( A  =/=  (/)  /\  A  C_  U. A
)  ->  om  ~<_  A ) ) )
9 eqid 2467 . . . 4  |-  ( y  e.  _V  |->  { w  e.  x  |  (
w  i^i  x )  C_  y } )  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
10 eqid 2467 . . . 4  |-  ( rec ( ( y  e. 
_V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } ) ,  (/) )  |`  om )  =  ( rec ( ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x
)  C_  y }
) ,  (/) )  |`  om )
119, 10, 1, 1inf3lem6 8060 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( rec ( ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x
)  C_  y }
) ,  (/) )  |`  om ) : om -1-1-> ~P x )
12 vex 3121 . . . . 5  |-  x  e. 
_V
1312pwex 4635 . . . 4  |-  ~P x  e.  _V
1413f1dom 7547 . . 3  |-  ( ( rec ( ( y  e.  _V  |->  { w  e.  x  |  (
w  i^i  x )  C_  y } ) ,  (/) )  |`  om ) : om -1-1-> ~P x  ->  om  ~<_  ~P x
)
15 pwfi 7825 . . . . . . 7  |-  ( x  e.  Fin  <->  ~P x  e.  Fin )
1615biimpi 194 . . . . . 6  |-  ( x  e.  Fin  ->  ~P x  e.  Fin )
17 isfinite 8079 . . . . . 6  |-  ( x  e.  Fin  <->  x  ~<  om )
18 isfinite 8079 . . . . . 6  |-  ( ~P x  e.  Fin  <->  ~P x  ~<  om )
1916, 17, 183imtr3i 265 . . . . 5  |-  ( x 
~<  om  ->  ~P x  ~<  om )
2019con3i 135 . . . 4  |-  ( -. 
~P x  ~<  om  ->  -.  x  ~<  om )
2113domtriom 8833 . . . 4  |-  ( om  ~<_  ~P x  <->  -.  ~P x  ~<  om )
2212domtriom 8833 . . . 4  |-  ( om  ~<_  x  <->  -.  x  ~<  om )
2320, 21, 223imtr4i 266 . . 3  |-  ( om  ~<_  ~P x  ->  om  ~<_  x )
2411, 14, 233syl 20 . 2  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om 
~<_  x )
251, 8, 24vtocl 3170 1  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4015   U.cuni 4250   class class class wbr 4452    |-> cmpt 4510    |` cres 5006   -1-1->wf1 5590   omcom 6694   reccrdg 7085    ~<_ cdom 7524    ~< csdm 7525   Fincfn 7526
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-reg 8028  ax-inf2 8068  ax-cc 8825
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-cda 8558
This theorem is referenced by:  axgroth3  9219
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