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Theorem isinffi 9407
 Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8717 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 9376 . . 3 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2 isinf 8717 . . 3 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎))
3 breq2 5034 . . . . . 6 (𝑎 = (card‘𝐵) → (𝑐𝑎𝑐 ≈ (card‘𝐵)))
43anbi2d 631 . . . . 5 (𝑎 = (card‘𝐵) → ((𝑐𝐴𝑐𝑎) ↔ (𝑐𝐴𝑐 ≈ (card‘𝐵))))
54exbidv 1922 . . . 4 (𝑎 = (card‘𝐵) → (∃𝑐(𝑐𝐴𝑐𝑎) ↔ ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵))))
65rspcva 3569 . . 3 (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐𝐴𝑐𝑎)) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
71, 2, 6syl2anr 599 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐𝐴𝑐 ≈ (card‘𝐵)))
8 simprr 772 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9 ficardid 9377 . . . . . . 7 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
109ad2antlr 726 . . . . . 6 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11 entr 8546 . . . . . 6 ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐𝐵)
128, 10, 11syl2anc 587 . . . . 5 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝑐𝐵)
1312ensymd 8545 . . . 4 (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) → 𝐵𝑐)
14 bren 8503 . . . 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 6589 . . . . . 6 (𝑓:𝐵1-1-onto𝑐𝑓:𝐵1-1𝑐)
17 simplrl 776 . . . . . 6 ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐𝐴𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵1-1-onto𝑐) → 𝑐𝐴)
18 f1ss 6555 . . . . . 6 ((𝑓:𝐵1-1𝑐𝑐𝐴) → 𝑓:𝐵1-1𝐴)
1916, 17, 18syl2an2 685 . . . . 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 1918 . . 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 1936 1 ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881   class class class wbr 5030  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324  ωcom 7562   ≈ cen 8491  Fincfn 8494  cardccrd 9350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7563  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-card 9354 This theorem is referenced by:  fidomtri  9408  hashdom  13738  erdsze2lem1  32575  eldioph2lem2  39717
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