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 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 7110	. . . 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 1865	. 2 ⊢ (𝐴 ∈ Fin → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ Fin))
4	3	con3dimp 638	1 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∃wex 1538   ∈ 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)