MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dominf Structured version   Unicode version

Theorem dominf 8828
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8818. See dominfac 8951 for a version proved from ax-ac 8842. The axiom of Regularity is used for this proof, via inf3lem6 8053, 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 2724 . . . 4  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 unieq 4242 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
53, 4sseq12d 3518 . . . 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 4441 . . 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 8053 . . 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 3098 . . . . 5  |-  x  e. 
_V
1312pwex 4620 . . . 4  |-  ~P x  e.  _V
1413f1dom 7539 . . 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 7817 . . . . . . 7  |-  ( x  e.  Fin  <->  ~P x  e.  Fin )
1615biimpi 194 . . . . . 6  |-  ( x  e.  Fin  ->  ~P x  e.  Fin )
17 isfinite 8072 . . . . . 6  |-  ( x  e.  Fin  <->  x  ~<  om )
18 isfinite 8072 . . . . . 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 8826 . . . 4  |-  ( om  ~<_  ~P x  <->  -.  ~P x  ~<  om )
2212domtriom 8826 . . . 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 3147 1  |-  ( ( A  =/=  (/)  /\  A  C_ 
U. A )  ->  om 
~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   U.cuni 4234   class class class wbr 4437    |-> cmpt 4495    |` cres 4991   -1-1->wf1 5575   omcom 6685   reccrdg 7077    ~<_ cdom 7516    ~< csdm 7517   Fincfn 7518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061  ax-cc 8818
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551
This theorem is referenced by:  axgroth3  9212
  Copyright terms: Public domain W3C validator