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Mirrors > Home > MPE Home > Th. List > rabssnn0fi | Structured version Visualization version GIF version |
Description: A subset of the nonnegative integers defined by a restricted class abstraction is finite if there is a nonnegative integer so that for all integers greater than this integer the condition of the class abstraction is not fulfilled. (Contributed by AV, 3-Oct-2019.) |
Ref | Expression |
---|---|
rabssnn0fi | ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4041 | . 2 ⊢ {𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 | |
2 | ssnn0fi 13899 | . . 3 ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 → ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}))) | |
3 | nnel 3055 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑}) | |
4 | nfcv 2904 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑦 | |
5 | nfcv 2904 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥ℕ0 | |
6 | nfsbc1v 3763 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑦 / 𝑥] ¬ 𝜑 | |
7 | 6 | nfn 1861 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥] ¬ 𝜑 |
8 | sbceq2a 3755 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜑)) | |
9 | 8 | equcoms 2024 | . . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜑)) |
10 | 9 | con2bid 355 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
11 | 4, 5, 7, 10 | elrabf 3645 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ (𝑦 ∈ ℕ0 ∧ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
12 | 11 | baib 537 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → (𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
13 | 3, 12 | bitrid 283 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (¬ 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
14 | 13 | con4bid 317 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ [𝑦 / 𝑥] ¬ 𝜑)) |
15 | 14 | imbi2d 341 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑))) |
16 | 15 | ralbiia 3091 | . . . . . 6 ⊢ (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ ∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑)) |
17 | nfv 1918 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑠 < 𝑦 | |
18 | 17, 6 | nfim 1900 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) |
19 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑠 < 𝑥 → ¬ 𝜑) | |
20 | breq2 5113 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥)) | |
21 | 20, 8 | imbi12d 345 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑠 < 𝑥 → ¬ 𝜑))) |
22 | 18, 19, 21 | cbvralw 3288 | . . . . . 6 ⊢ (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) |
23 | 16, 22 | bitri 275 | . . . . 5 ⊢ (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) |
24 | 23 | a1i 11 | . . . 4 ⊢ (({𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0) → (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))) |
25 | 24 | rexbidva 3170 | . . 3 ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 → (∃𝑠 ∈ ℕ0 ∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))) |
26 | 2, 25 | bitrd 279 | . 2 ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 → ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))) |
27 | 1, 26 | ax-mp 5 | 1 ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∉ wnel 3046 ∀wral 3061 ∃wrex 3070 {crab 3406 [wsbc 3743 ⊆ wss 3914 class class class wbr 5109 Fincfn 8889 < clt 11197 ℕ0cn0 12421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: fsuppmapnn0ub 13909 mptnn0fsupp 13911 mptnn0fsuppr 13913 pmatcollpw2lem 22149 |
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