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Mirrors > Home > MPE Home > Th. List > fiinfnf1o | Structured version Visualization version GIF version |
Description: There is no bijection between a finite set and an infinite set. (Contributed by Alexander van der Vekens, 25-Dec-2017.) |
Ref | Expression |
---|---|
fiinfnf1o | ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofo 6792 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
2 | fofi 9285 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ Fin) | |
3 | 2 | ex 414 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ Fin)) |
4 | 1, 3 | syl5 34 | . . 3 ⊢ (𝐴 ∈ Fin → (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ Fin)) |
5 | 4 | exlimdv 1937 | . 2 ⊢ (𝐴 ∈ Fin → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ Fin)) |
6 | 5 | con3dimp 410 | 1 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 –onto→wfo 6495 –1-1-onto→wf1o 6496 Fincfn 8886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-fin 8890 |
This theorem is referenced by: hasheqf1oi 14257 |
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