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Theorem isinffi 9000
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8330 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
isinffi ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem isinffi
Dummy variables 𝑐 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ficardom 8969 . . 3 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2 isinf 8330 . . 3 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎))
3 breq2 4800 . . . . . 6 (𝑎 = (card‘𝐵) → (𝑐𝑎𝑐 ≈ (card‘𝐵)))
43anbi2d 742 . . . . 5 (𝑎 = (card‘𝐵) → ((𝑐𝐴𝑐𝑎) ↔ (𝑐𝐴𝑐 ≈ (card‘𝐵))))
54exbidv 1991 . . . 4 (𝑎 = (card‘𝐵) → (∃𝑐(𝑐𝐴𝑐𝑎) ↔ ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵))))
65rspcva 3439 . . 3 (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎)) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
71, 2, 6syl2anr 496 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
8 simprr 813 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9 ficardid 8970 . . . . . . 7 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
109ad2antlr 765 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11 entr 8165 . . . . . 6 ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐𝐵)
128, 10, 11syl2anc 696 . . . . 5 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐𝐵)
1312ensymd 8164 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝐵𝑐)
14 bren 8122 . . . 4 (𝐵𝑐 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑐)
1513, 14sylib 208 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1-onto𝑐)
16 f1of1 6289 . . . . . . 7 (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝑐)
1716adantl 473 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝑐)
18 simplrl 819 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑐𝐴)
19 f1ss 6259 . . . . . 6 ((𝑓:𝐵1-1𝑐𝑐𝐴) → 𝑓:𝐵1-1𝐴)
2017, 18, 19syl2anc 696 . . . . 5 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝐴)
2120ex 449 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝐴))
2221eximdv 1987 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵1-1-onto𝑐 → ∃𝑓 𝑓:𝐵1-1𝐴))
2315, 22mpd 15 . 2 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1𝐴)
247, 23exlimddv 2004 1 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1624  wex 1845  wcel 2131  wral 3042  wss 3707   class class class wbr 4796  1-1wf1 6038  1-1-ontowf1o 6040  cfv 6041  ωcom 7222  cen 8110  Fincfn 8113  cardccrd 8943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947
This theorem is referenced by:  fidomtri  9001  hashdom  13352  erdsze2lem1  31484  eldioph2lem2  37818
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