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Mirrors > Home > MPE Home > Th. List > ishashinf | Structured version Visualization version GIF version |
Description: Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8058. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
Ref | Expression |
---|---|
ishashinf | ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 12634 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1...𝑛) ∈ Fin) | |
2 | ficardom 8670 | . . . . . 6 ⊢ ((1...𝑛) ∈ Fin → (card‘(1...𝑛)) ∈ ω) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (card‘(1...𝑛)) ∈ ω) |
4 | isinf 8058 | . . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑎)) | |
5 | breq2 4587 | . . . . . . . 8 ⊢ (𝑎 = (card‘(1...𝑛)) → (𝑥 ≈ 𝑎 ↔ 𝑥 ≈ (card‘(1...𝑛)))) | |
6 | 5 | anbi2d 736 | . . . . . . 7 ⊢ (𝑎 = (card‘(1...𝑛)) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑎) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛))))) |
7 | 6 | exbidv 1837 | . . . . . 6 ⊢ (𝑎 = (card‘(1...𝑛)) → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑎) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛))))) |
8 | 7 | rspcva 3280 | . . . . 5 ⊢ (((card‘(1...𝑛)) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑎)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛)))) |
9 | 3, 4, 8 | syl2anr 494 | . . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛)))) |
10 | selpw 4115 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
11 | 10 | biimpri 217 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝒫 𝐴) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) |
13 | hasheni 12998 | . . . . . . . . 9 ⊢ (𝑥 ≈ (card‘(1...𝑛)) → (#‘𝑥) = (#‘(card‘(1...𝑛)))) | |
14 | 13 | adantl 481 | . . . . . . . 8 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘𝑥) = (#‘(card‘(1...𝑛)))) |
15 | hashcard 13008 | . . . . . . . . . . 11 ⊢ ((1...𝑛) ∈ Fin → (#‘(card‘(1...𝑛))) = (#‘(1...𝑛))) | |
16 | 1, 15 | syl 17 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (#‘(card‘(1...𝑛))) = (#‘(1...𝑛))) |
17 | nnnn0 11176 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
18 | hashfz1 12996 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛) | |
19 | 17, 18 | syl 17 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (#‘(1...𝑛)) = 𝑛) |
20 | 16, 19 | eqtrd 2644 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (#‘(card‘(1...𝑛))) = 𝑛) |
21 | 20 | ad2antlr 759 | . . . . . . . 8 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘(card‘(1...𝑛))) = 𝑛) |
22 | 14, 21 | eqtrd 2644 | . . . . . . 7 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≈ (card‘(1...𝑛))) → (#‘𝑥) = 𝑛) |
23 | 22 | ex 449 | . . . . . 6 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (𝑥 ≈ (card‘(1...𝑛)) → (#‘𝑥) = 𝑛)) |
24 | 12, 23 | anim12d 584 | . . . . 5 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛))) → (𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛))) |
25 | 24 | eximdv 1833 | . . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (card‘(1...𝑛))) → ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛))) |
26 | 9, 25 | mpd 15 | . . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛)) |
27 | df-rex 2902 | . . 3 ⊢ (∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝑛)) | |
28 | 26, 27 | sylibr 223 | . 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛) |
29 | 28 | ralrimiva 2949 | 1 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≈ cen 7838 Fincfn 7841 cardccrd 8644 1c1 9816 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: esumcst 29452 sge0rpcpnf 39314 |
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