Theorem fiinfnf1o 11203

Description: There is no bijection between a finite set and an infinite set. By infnfi 7189 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:𝐴–1-1-onto→𝐵)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fiinfnf1o

Step	Hyp	Ref	Expression
1		f1ofi 7247	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐵 ∈ Fin)
2	1	ex 115	. . 3 ⊢ (𝐴 ∈ Fin → (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ Fin))
3	2	exlimdv 1872	. 2 ⊢ (𝐴 ∈ Fin → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ Fin))
4	3	con3dimp 644	1 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∃wex 1545   ∈ wcel 2209  –1-1-onto→wf1o 5371  Fincfn 7012

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-er 6797  df-en 7013  df-fin 7015

This theorem is referenced by: (None)