ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fiinfnf1o Unicode version

Theorem fiinfnf1o 10539
Description: There is no bijection between a finite set and an infinite set. By infnfi 6789 the theorem would also hold if "infinite" were expressed as  om  ~<_  B. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
Assertion
Ref Expression
fiinfnf1o  |-  ( ( A  e.  Fin  /\  -.  B  e.  Fin )  ->  -.  E. f 
f : A -1-1-onto-> B )
Distinct variable groups:    A, f    B, f

Proof of Theorem fiinfnf1o
StepHypRef Expression
1 f1ofi 6831 . . . 4  |-  ( ( A  e.  Fin  /\  f : A -1-1-onto-> B )  ->  B  e.  Fin )
21ex 114 . . 3  |-  ( A  e.  Fin  ->  (
f : A -1-1-onto-> B  ->  B  e.  Fin )
)
32exlimdv 1791 . 2  |-  ( A  e.  Fin  ->  ( E. f  f : A
-1-1-onto-> B  ->  B  e.  Fin ) )
43con3dimp 624 1  |-  ( ( A  e.  Fin  /\  -.  B  e.  Fin )  ->  -.  E. f 
f : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   E.wex 1468    e. wcel 1480   -1-1-onto->wf1o 5122   Fincfn 6634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-er 6429  df-en 6635  df-fin 6637
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator