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Theorem dominfac 8742
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8633. See dominf 8619 for a version proved from ax-cc 8609. (Contributed by NM, 25-Mar-2007.)
Hypothesis
Ref Expression
dominfac.1  |-  A  e. 
_V
Assertion
Ref Expression
dominfac  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )

Proof of Theorem dominfac
Dummy variables  x  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dominfac.1 . 2  |-  A  e. 
_V
2 neeq1 2621 . . . 4  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 unieq 4104 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
53, 4sseq12d 3390 . . . 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 4301 . . 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 2443 . . . 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 2443 . . . 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 7844 . . 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 2980 . . . . 5  |-  x  e. 
_V
1312pwex 4480 . . . 4  |-  ~P x  e.  _V
1413f1dom 7336 . . 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 7611 . . . . . . 7  |-  ( x  e.  Fin  <->  ~P x  e.  Fin )
1615biimpi 194 . . . . . 6  |-  ( x  e.  Fin  ->  ~P x  e.  Fin )
17 isfinite 7863 . . . . . 6  |-  ( x  e.  Fin  <->  x  ~<  om )
18 isfinite 7863 . . . . . 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 )
21 omex 7854 . . . . 5  |-  om  e.  _V
22 domtri 8725 . . . . 5  |-  ( ( om  e.  _V  /\  ~P x  e.  _V )  ->  ( om  ~<_  ~P x  <->  -. 
~P x  ~<  om )
)
2321, 13, 22mp2an 672 . . . 4  |-  ( om  ~<_  ~P x  <->  -.  ~P x  ~<  om )
24 domtri 8725 . . . . 5  |-  ( ( om  e.  _V  /\  x  e.  _V )  ->  ( om  ~<_  x  <->  -.  x  ~<  om ) )
2521, 12, 24mp2an 672 . . . 4  |-  ( om  ~<_  x  <->  -.  x  ~<  om )
2620, 23, 253imtr4i 266 . . 3  |-  ( om  ~<_  ~P x  ->  om  ~<_  x )
2711, 14, 263syl 20 . 2  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om 
~<_  x )
281, 8, 27vtocl 3029 1  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   {crab 2724   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297    e. cmpt 4355    |` cres 4847   -1-1->wf1 5420   omcom 6481   reccrdg 6870    ~<_ cdom 7313    ~< csdm 7314   Fincfn 7315
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-reg 7812  ax-inf2 7852  ax-ac2 8637
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-ac 8291
This theorem is referenced by: (None)
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