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Mirrors > Home > MPE Home > Th. List > nn1m1nn | Structured version Visualization version GIF version |
Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1m1nn | ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 863 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ)) | |
2 | 1cnd 10630 | . . 3 ⊢ (𝑥 = 1 → 1 ∈ ℂ) | |
3 | 1, 2 | 2thd 267 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ)) |
4 | eqeq1 2825 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | oveq1 7157 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1)) | |
6 | 5 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ)) |
7 | 4, 6 | orbi12d 915 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ))) |
8 | eqeq1 2825 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
9 | oveq1 7157 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1)) | |
10 | 9 | eleq1d 2897 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ)) |
11 | 8, 10 | orbi12d 915 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
12 | eqeq1 2825 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
13 | oveq1 7157 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1)) | |
14 | 13 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ)) |
15 | 12, 14 | orbi12d 915 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))) |
16 | ax-1cn 10589 | . 2 ⊢ 1 ∈ ℂ | |
17 | nncn 11640 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | pncan 10886 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦) | |
19 | 17, 16, 18 | sylancl 588 | . . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦) |
20 | id 22 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
21 | 19, 20 | eqeltrd 2913 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ) |
22 | 21 | olcd 870 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ)) |
23 | 22 | a1d 25 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ) → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
24 | 3, 7, 11, 15, 16, 23 | nnind 11650 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 − cmin 10864 ℕcn 11632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-nn 11633 |
This theorem is referenced by: nn1suc 11653 nnsub 11675 nnm1nn0 11932 nn0ge2m1nn 11958 elfznelfzo 13136 psgnfzto1stlem 30737 ballotlemfc0 31745 ballotlemfcc 31746 stirlinglem5 42357 |
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