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

Theorem infpwfidom 7671
Description: The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
infpwfidom  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )

Proof of Theorem infpwfidom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snelpwi 4236 . . 3  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
2 snfi 6957 . . . 4  |-  { x }  e.  Fin
32a1i 10 . . 3  |-  ( x  e.  A  ->  { x }  e.  Fin )
4 elin 3371 . . 3  |-  ( { x }  e.  ( ~P A  i^i  Fin ) 
<->  ( { x }  e.  ~P A  /\  {
x }  e.  Fin ) )
51, 3, 4sylanbrc 645 . 2  |-  ( x  e.  A  ->  { x }  e.  ( ~P A  i^i  Fin ) )
6 sneqbg 3799 . . 3  |-  ( x  e.  A  ->  ( { x }  =  { y }  <->  x  =  y ) )
76adantr 451 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
85, 7dom2 6920 1  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   ~Pcpw 3638   {csn 3653   class class class wbr 4039    ~<_ cdom 6877   Fincfn 6879
This theorem is referenced by:  infpwfien  7705  ttukeylem1  8152  canthnum  8287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-en 6880  df-dom 6881  df-fin 6883
  Copyright terms: Public domain W3C validator