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Theorem fiinfnf1o 10043
Description: There is no bijection between a finite set and an infinite set. By infnfi 6544 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 6580 . . . 4 ((𝐴 ∈ Fin ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐵 ∈ Fin)
21ex 113 . . 3 (𝐴 ∈ Fin → (𝑓:𝐴1-1-onto𝐵𝐵 ∈ Fin))
32exlimdv 1744 . 2 (𝐴 ∈ Fin → (∃𝑓 𝑓:𝐴1-1-onto𝐵𝐵 ∈ Fin))
43con3dimp 597 1 ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wex 1424  wcel 1436  1-1-ontowf1o 4971  Fincfn 6390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-er 6225  df-en 6391  df-fin 6393
This theorem is referenced by: (None)
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