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Theorem ishashinf 13182
Description: Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8118. (Contributed by Thierry Arnoux, 5-Jul-2017.)
Assertion
Ref Expression
ishashinf 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)
Distinct variable group:   𝑥,𝑛,𝐴

Proof of Theorem ishashinf
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fzfid 12709 . . . . . 6 (𝑛 ∈ ℕ → (1...𝑛) ∈ Fin)
2 ficardom 8732 . . . . . 6 ((1...𝑛) ∈ Fin → (card‘(1...𝑛)) ∈ ω)
31, 2syl 17 . . . . 5 (𝑛 ∈ ℕ → (card‘(1...𝑛)) ∈ ω)
4 isinf 8118 . . . . 5 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑎))
5 breq2 4622 . . . . . . . 8 (𝑎 = (card‘(1...𝑛)) → (𝑥𝑎𝑥 ≈ (card‘(1...𝑛))))
65anbi2d 739 . . . . . . 7 (𝑎 = (card‘(1...𝑛)) → ((𝑥𝐴𝑥𝑎) ↔ (𝑥𝐴𝑥 ≈ (card‘(1...𝑛)))))
76exbidv 1852 . . . . . 6 (𝑎 = (card‘(1...𝑛)) → (∃𝑥(𝑥𝐴𝑥𝑎) ↔ ∃𝑥(𝑥𝐴𝑥 ≈ (card‘(1...𝑛)))))
87rspcva 3298 . . . . 5 (((card‘(1...𝑛)) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑎)) → ∃𝑥(𝑥𝐴𝑥 ≈ (card‘(1...𝑛))))
93, 4, 8syl2anr 495 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥𝐴𝑥 ≈ (card‘(1...𝑛))))
10 selpw 4142 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110biimpri 218 . . . . . . 7 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
1211a1i 11 . . . . . 6 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
13 hasheni 13073 . . . . . . . . 9 (𝑥 ≈ (card‘(1...𝑛)) → (#‘𝑥) = (#‘(card‘(1...𝑛))))
1413adantl 482 . . . . . . . 8 (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘𝑥) = (#‘(card‘(1...𝑛))))
15 hashcard 13083 . . . . . . . . . . 11 ((1...𝑛) ∈ Fin → (#‘(card‘(1...𝑛))) = (#‘(1...𝑛)))
161, 15syl 17 . . . . . . . . . 10 (𝑛 ∈ ℕ → (#‘(card‘(1...𝑛))) = (#‘(1...𝑛)))
17 nnnn0 11244 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
18 hashfz1 13071 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛)
1917, 18syl 17 . . . . . . . . . 10 (𝑛 ∈ ℕ → (#‘(1...𝑛)) = 𝑛)
2016, 19eqtrd 2660 . . . . . . . . 9 (𝑛 ∈ ℕ → (#‘(card‘(1...𝑛))) = 𝑛)
2120ad2antlr 762 . . . . . . . 8 (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘(card‘(1...𝑛))) = 𝑛)
2214, 21eqtrd 2660 . . . . . . 7 (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘𝑥) = 𝑛)
2322ex 450 . . . . . 6 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (𝑥 ≈ (card‘(1...𝑛)) → (#‘𝑥) = 𝑛))
2412, 23anim12d 585 . . . . 5 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ((𝑥𝐴𝑥 ≈ (card‘(1...𝑛))) → (𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛)))
2524eximdv 1848 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (∃𝑥(𝑥𝐴𝑥 ≈ (card‘(1...𝑛))) → ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛)))
269, 25mpd 15 . . 3 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛))
27 df-rex 2918 . . 3 (∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛))
2826, 27sylibr 224 . 2 ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)
2928ralrimiva 2965 1 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1992  wral 2912  wrex 2913  wss 3560  𝒫 cpw 4135   class class class wbr 4618  cfv 5850  (class class class)co 6605  ωcom 7013  cen 7897  Fincfn 7900  cardccrd 8706  1c1 9882  cn 10965  0cn0 11237  ...cfz 12265  #chash 13054
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  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  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-iun 4492  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-pred 5642  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-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055
This theorem is referenced by:  esumcst  29898  sge0rpcpnf  39913
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