![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3828 | . 2 ⊢ {𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 | |
2 | ssnn0fi 12998 | . . 3 ⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ⊆ ℕ0 → ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}))) | |
3 | nnel 3044 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑}) | |
4 | nfcv 2902 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑦 | |
5 | nfcv 2902 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥ℕ0 | |
6 | nfsbc1v 3596 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑦 / 𝑥] ¬ 𝜑 | |
7 | 6 | nfn 1933 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥] ¬ 𝜑 |
8 | sbceq2a 3588 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜑)) | |
9 | 8 | equcoms 2102 | . . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜑)) |
10 | 9 | con2bid 343 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
11 | 4, 5, 7, 10 | elrabf 3500 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ (𝑦 ∈ ℕ0 ∧ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
12 | 11 | baib 982 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → (𝑦 ∈ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
13 | 3, 12 | syl5bb 272 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (¬ 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥] ¬ 𝜑)) |
14 | 13 | con4bid 306 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑} ↔ [𝑦 / 𝑥] ¬ 𝜑)) |
15 | 14 | imbi2d 329 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑))) |
16 | 15 | ralbiia 3117 | . . . . . 6 ⊢ (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → 𝑦 ∉ {𝑥 ∈ ℕ0 ∣ 𝜑}) ↔ ∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑)) |
17 | nfv 1992 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑠 < 𝑦 | |
18 | 17, 6 | nfim 1974 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) |
19 | nfv 1992 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑠 < 𝑥 → ¬ 𝜑) | |
20 | breq2 4808 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑠 < 𝑦 ↔ 𝑠 < 𝑥)) | |
21 | 20, 8 | imbi12d 333 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑠 < 𝑥 → ¬ 𝜑))) |
22 | 18, 19, 21 | cbvral 3306 | . . . . . 6 ⊢ (∀𝑦 ∈ ℕ0 (𝑠 < 𝑦 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) |
23 | 16, 22 | bitri 264 | . . . . 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 3187 | . . 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 268 | . 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 196 ∧ wa 383 ∈ wcel 2139 ∉ wnel 3035 ∀wral 3050 ∃wrex 3051 {crab 3054 [wsbc 3576 ⊆ wss 3715 class class class wbr 4804 Fincfn 8123 < clt 10286 ℕ0cn0 11504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 |
This theorem is referenced by: fsuppmapnn0ub 13009 mptnn0fsupp 13011 mptnn0fsuppr 13013 pmatcollpw2lem 20804 |
Copyright terms: Public domain | W3C validator |