Theorem fiinfnf1o 11008

Description: There is no bijection between a finite set and an infinite set. By infnfi 7057 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

Step	Hyp	Ref	Expression
1		f1ofi 7110	. . . 4 |- ( ( A e. Fin /\ f : A -1-1-onto-> B ) -> B e. Fin )
2	1	ex 115	. . 3 |- ( A e. Fin -> ( f : A -1-1-onto-> B -> B e. Fin ) )
3	2	exlimdv 1865	. 2 |- ( A e. Fin -> ( E. f f : A -1-1-onto-> B -> B e. Fin ) )
4	3	con3dimp 638	1 |- ( ( A e. Fin /\ -. B e. Fin ) -> -. E. f f : A -1-1-onto-> B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   E.wex 1538   e. wcel 2200   -1-1-onto->wf1o 5317   Fincfn 6887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-er 6680  df-en 6888  df-fin 6890

This theorem is referenced by: (None)