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Theorem isinffi 8763
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8118 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 8732 . . 3 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2 isinf 8118 . . 3 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎))
3 breq2 4622 . . . . . 6 (𝑎 = (card‘𝐵) → (𝑐𝑎𝑐 ≈ (card‘𝐵)))
43anbi2d 739 . . . . 5 (𝑎 = (card‘𝐵) → ((𝑐𝐴𝑐𝑎) ↔ (𝑐𝐴𝑐 ≈ (card‘𝐵))))
54exbidv 1852 . . . 4 (𝑎 = (card‘𝐵) → (∃𝑐(𝑐𝐴𝑐𝑎) ↔ ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵))))
65rspcva 3298 . . 3 (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎)) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
71, 2, 6syl2anr 495 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
8 simprr 795 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9 ficardid 8733 . . . . . . 7 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
109ad2antlr 762 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11 entr 7953 . . . . . 6 ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐𝐵)
128, 10, 11syl2anc 692 . . . . 5 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐𝐵)
1312ensymd 7952 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝐵𝑐)
14 bren 7909 . . . 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 6095 . . . . . . 7 (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝑐)
1716adantl 482 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝑐)
18 simplrl 799 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑐𝐴)
19 f1ss 6065 . . . . . 6 ((𝑓:𝐵1-1𝑐𝑐𝐴) → 𝑓:𝐵1-1𝐴)
2017, 18, 19syl2anc 692 . . . . 5 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝐴)
2120ex 450 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝐴))
2221eximdv 1848 . . 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 1865 1 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1992  wral 2912  wss 3560   class class class wbr 4618  1-1wf1 5847  1-1-ontowf1o 5849  cfv 5850  ωcom 7013  cen 7897  Fincfn 7900  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710
This theorem is referenced by:  fidomtri  8764  hashdom  13105  erdsze2lem1  30885  eldioph2lem2  36790
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