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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 13933 | . 2 ⊢ < We (ℤ≥‘𝑀) | |
2 | fvex 6904 | . . 3 ⊢ (ℤ≥‘𝑀) ∈ V | |
3 | exse 5639 | . . 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 13630 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → Pred( < , (ℤ≥‘𝑀), 𝑥) = (𝑀...(𝑥 − 1))) | |
8 | 7 | raleqdv 3324 | . . 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 6362 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 Se wse 5629 Predcpred 6299 ‘cfv 6543 (class class class)co 7412 1c1 11117 < clt 11255 − cmin 11451 ℤ≥cuz 12829 ...cfz 13491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 |
This theorem is referenced by: nnsinds 13960 nn0sinds 13961 |
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