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Theorem fiinfnf1o 10759
Description: There is no bijection between a finite set and an infinite set. By infnfi 6892 the theorem would also hold if "infinite" were expressed as ω ≼ 𝐵. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
Assertion
Ref Expression
fiinfnf1o ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fiinfnf1o
StepHypRef Expression
1 f1ofi 6939 . . . 4 ((𝐴 ∈ Fin ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐵 ∈ Fin)
21ex 115 . . 3 (𝐴 ∈ Fin → (𝑓:𝐴1-1-onto𝐵𝐵 ∈ Fin))
32exlimdv 1819 . 2 (𝐴 ∈ Fin → (∃𝑓 𝑓:𝐴1-1-onto𝐵𝐵 ∈ Fin))
43con3dimp 635 1 ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wex 1492  wcel 2148  1-1-ontowf1o 5214  Fincfn 6737
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-er 6532  df-en 6738  df-fin 6740
This theorem is referenced by: (None)
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