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Mirrors > Home > MPE Home > Th. List > peano5uzti | Structured version Visualization version GIF version |
Description: Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.) |
Ref | Expression |
---|---|
peano5uzti | ⊢ (𝑁 ∈ ℤ → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . . . 4 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → (𝑁 ∈ 𝐴 ↔ if(𝑁 ∈ ℤ, 𝑁, 1) ∈ 𝐴)) | |
2 | 1 | anbi1d 631 | . . 3 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) ↔ (if(𝑁 ∈ ℤ, 𝑁, 1) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴))) |
3 | breq1 5055 | . . . . 5 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → (𝑁 ≤ 𝑘 ↔ if(𝑁 ∈ ℤ, 𝑁, 1) ≤ 𝑘)) | |
4 | 3 | rabbidv 3472 | . . . 4 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ if(𝑁 ∈ ℤ, 𝑁, 1) ≤ 𝑘}) |
5 | 4 | sseq1d 3986 | . . 3 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → ({𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴 ↔ {𝑘 ∈ ℤ ∣ if(𝑁 ∈ ℤ, 𝑁, 1) ≤ 𝑘} ⊆ 𝐴)) |
6 | 2, 5 | imbi12d 347 | . 2 ⊢ (𝑁 = if(𝑁 ∈ ℤ, 𝑁, 1) → (((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴) ↔ ((if(𝑁 ∈ ℤ, 𝑁, 1) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ if(𝑁 ∈ ℤ, 𝑁, 1) ≤ 𝑘} ⊆ 𝐴))) |
7 | 1z 11999 | . . . 4 ⊢ 1 ∈ ℤ | |
8 | 7 | elimel 4520 | . . 3 ⊢ if(𝑁 ∈ ℤ, 𝑁, 1) ∈ ℤ |
9 | 8 | peano5uzi 12058 | . 2 ⊢ ((if(𝑁 ∈ ℤ, 𝑁, 1) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ if(𝑁 ∈ ℤ, 𝑁, 1) ≤ 𝑘} ⊆ 𝐴) |
10 | 6, 9 | dedth 4509 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ⊆ wss 3924 ifcif 4453 class class class wbr 5052 (class class class)co 7142 1c1 10524 + caddc 10526 ≤ cle 10662 ℤcz 11968 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-n0 11885 df-z 11969 |
This theorem is referenced by: uzind 12061 |
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