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Theorem isinffi 9608
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8891 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 9577 . . 3 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2 isinf 8891 . . 3 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎))
3 breq2 5057 . . . . . 6 (𝑎 = (card‘𝐵) → (𝑐𝑎𝑐 ≈ (card‘𝐵)))
43anbi2d 632 . . . . 5 (𝑎 = (card‘𝐵) → ((𝑐𝐴𝑐𝑎) ↔ (𝑐𝐴𝑐 ≈ (card‘𝐵))))
54exbidv 1929 . . . 4 (𝑎 = (card‘𝐵) → (∃𝑐(𝑐𝐴𝑐𝑎) ↔ ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵))))
65rspcva 3535 . . 3 (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎)) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
71, 2, 6syl2anr 600 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
8 simprr 773 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9 ficardid 9578 . . . . . . 7 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
109ad2antlr 727 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11 entr 8680 . . . . . 6 ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐𝐵)
128, 10, 11syl2anc 587 . . . . 5 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐𝐵)
1312ensymd 8679 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝐵𝑐)
14 bren 8636 . . . 4 (𝐵𝑐 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑐)
1513, 14sylib 221 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1-onto𝑐)
16 f1of1 6660 . . . . . 6 (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝑐)
17 simplrl 777 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑐𝐴)
18 f1ss 6621 . . . . . 6 ((𝑓:𝐵1-1𝑐𝑐𝐴) → 𝑓:𝐵1-1𝐴)
1916, 17, 18syl2an2 686 . . . . 5 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑓:𝐵1-1𝐴)
2019ex 416 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝐴))
2120eximdv 1925 . . 3 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵1-1-onto𝑐 → ∃𝑓 𝑓:𝐵1-1𝐴))
2215, 21mpd 15 . 2 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵1-1𝐴)
237, 22exlimddv 1943 1 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wex 1787  wcel 2110  wral 3061  wss 3866   class class class wbr 5053  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  ωcom 7644  cen 8623  Fincfn 8626  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555
This theorem is referenced by:  fidomtri  9609  hashdom  13946  erdsze2lem1  32878  eldioph2lem2  40286
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