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Theorem dominf 8816
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8806. See dominfac 8939 for a version proved from ax-ac 8830. The axiom of Regularity is used for this proof, via inf3lem6 8041, 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 2735 . . . 4  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 unieq 4243 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
53, 4sseq12d 3518 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. x  <->  A 
C_  U. A ) )
62, 5anbi12d 708 . . 3  |-  ( x  =  A  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  <->  ( A  =/=  (/)  /\  A  C_  U. A ) ) )
7 breq2 4443 . . 3  |-  ( x  =  A  ->  ( om 
~<_  x  <->  om  ~<_  A ) )
86, 7imbi12d 318 . 2  |-  ( x  =  A  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  ~<_  x )  <-> 
( ( A  =/=  (/)  /\  A  C_  U. A
)  ->  om  ~<_  A ) ) )
9 eqid 2454 . . . 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 2454 . . . 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 8041 . . 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 3109 . . . . 5  |-  x  e. 
_V
1312pwex 4620 . . . 4  |-  ~P x  e.  _V
1413f1dom 7530 . . 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 7807 . . . . . . 7  |-  ( x  e.  Fin  <->  ~P x  e.  Fin )
1615biimpi 194 . . . . . 6  |-  ( x  e.  Fin  ->  ~P x  e.  Fin )
17 isfinite 8060 . . . . . 6  |-  ( x  e.  Fin  <->  x  ~<  om )
18 isfinite 8060 . . . . . 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 8814 . . . 4  |-  ( om  ~<_  ~P x  <->  -.  ~P x  ~<  om )
2212domtriom 8814 . . . 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 3158 1  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497    |` cres 4990   -1-1->wf1 5567   omcom 6673   reccrdg 7067    ~<_ cdom 7507    ~< csdm 7508   Fincfn 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049  ax-cc 8806
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539
This theorem is referenced by:  axgroth3  9198
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