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Mirrors > Home > MPE Home > Th. List > uzsinds | Structured version Visualization version GIF version |
Description: Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
uzsinds.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
uzsinds.2 | ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) |
uzsinds.3 | ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)) |
Ref | Expression |
---|---|
uzsinds | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltweuz 13084 | . 2 ⊢ < We (ℤ≥‘𝑀) | |
2 | fvex 6461 | . . 3 ⊢ (ℤ≥‘𝑀) ∈ V | |
3 | exse 5321 | . . 3 ⊢ ((ℤ≥‘𝑀) ∈ V → < Se (ℤ≥‘𝑀)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ < Se (ℤ≥‘𝑀) |
5 | uzsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | uzsinds.2 | . 2 ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) | |
7 | preduz 12785 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → Pred( < , (ℤ≥‘𝑀), 𝑥) = (𝑀...(𝑥 − 1))) | |
8 | 7 | raleqdv 3340 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ Pred ( < , (ℤ≥‘𝑀), 𝑥)𝜓 ↔ ∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓)) |
9 | uzsinds.3 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)) | |
10 | 8, 9 | sylbid 232 | . 2 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ Pred ( < , (ℤ≥‘𝑀), 𝑥)𝜓 → 𝜑)) |
11 | 1, 4, 5, 6, 10 | wfis3 5976 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 Se wse 5314 Predcpred 5934 ‘cfv 6137 (class class class)co 6924 1c1 10275 < clt 10413 − cmin 10608 ℤ≥cuz 11997 ...cfz 12648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 |
This theorem is referenced by: nnsinds 13111 nn0sinds 13112 |
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