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Theorem fiinfnf1o 10500
Description: There is no bijection between a finite set and an infinite set. By infnfi 6757 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 6799 . . . 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 1775 . 2  |-  ( A  e.  Fin  ->  ( E. f  f : A
-1-1-onto-> B  ->  B  e.  Fin ) )
43con3dimp 609 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 1453    e. wcel 1465   -1-1-onto->wf1o 5092   Fincfn 6602
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by: (None)
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