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Theorem dominf 8289
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8279. See dominfac 8412 for a version proved from ax-ac 8303. The axiom of Regularity is used for this proof, via inf3lem6 7552, 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 2583 . . . 4  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
3 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 unieq 3992 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
53, 4sseq12d 3345 . . . 4  |-  ( x  =  A  ->  (
x  C_  U. x  <->  A 
C_  U. A ) )
62, 5anbi12d 692 . . 3  |-  ( x  =  A  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  <->  ( A  =/=  (/)  /\  A  C_  U. A ) ) )
7 breq2 4184 . . 3  |-  ( x  =  A  ->  ( om 
~<_  x  <->  om  ~<_  A ) )
86, 7imbi12d 312 . 2  |-  ( x  =  A  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  ~<_  x )  <-> 
( ( A  =/=  (/)  /\  A  C_  U. A
)  ->  om  ~<_  A ) ) )
9 eqid 2412 . . . 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 2412 . . . 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 7552 . . 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 2927 . . . . 5  |-  x  e. 
_V
1312pwex 4350 . . . 4  |-  ~P x  e.  _V
1413f1dom 7096 . . 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 7368 . . . . . . 7  |-  ( x  e.  Fin  <->  ~P x  e.  Fin )
1615biimpi 187 . . . . . 6  |-  ( x  e.  Fin  ->  ~P x  e.  Fin )
17 isfinite 7571 . . . . . 6  |-  ( x  e.  Fin  <->  x  ~<  om )
18 isfinite 7571 . . . . . 6  |-  ( ~P x  e.  Fin  <->  ~P x  ~<  om )
1916, 17, 183imtr3i 257 . . . . 5  |-  ( x 
~<  om  ->  ~P x  ~<  om )
2019con3i 129 . . . 4  |-  ( -. 
~P x  ~<  om  ->  -.  x  ~<  om )
2113domtriom 8287 . . . 4  |-  ( om  ~<_  ~P x  <->  -.  ~P x  ~<  om )
2212domtriom 8287 . . . 4  |-  ( om  ~<_  x  <->  -.  x  ~<  om )
2320, 21, 223imtr4i 258 . . 3  |-  ( om  ~<_  ~P x  ->  om  ~<_  x )
2411, 14, 233syl 19 . 2  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om 
~<_  x )
251, 8, 24vtocl 2974 1  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   {crab 2678   _Vcvv 2924    i^i cin 3287    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   U.cuni 3983   class class class wbr 4180    e. cmpt 4234   omcom 4812    |` cres 4847   -1-1->wf1 5418   reccrdg 6634    ~<_ cdom 7074    ~< csdm 7075   Fincfn 7076
This theorem is referenced by:  axgroth3  8670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-reg 7524  ax-inf2 7560  ax-cc 8279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-cda 8012
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