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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 13966 | . 2 ⊢ < We (ℤ≥‘𝑀) | |
2 | fvex 6915 | . . 3 ⊢ (ℤ≥‘𝑀) ∈ V | |
3 | exse 5645 | . . 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 13663 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → Pred( < , (ℤ≥‘𝑀), 𝑥) = (𝑀...(𝑥 − 1))) | |
8 | 7 | raleqdv 3323 | . . 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 239 | . 2 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ Pred ( < , (ℤ≥‘𝑀), 𝑥)𝜓 → 𝜑)) |
11 | 1, 4, 5, 6, 10 | wfis3 6372 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 Se wse 5635 Predcpred 6309 ‘cfv 6553 (class class class)co 7426 1c1 11147 < clt 11286 − cmin 11482 ℤ≥cuz 12860 ...cfz 13524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 |
This theorem is referenced by: nnsinds 13993 nn0sinds 13994 |
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