Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elnn0nn | Structured version Visualization version GIF version |
Description: The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
elnn0nn | ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 11910 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
2 | nn0p1nn 11939 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | jca 514 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
4 | simpl 485 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → 𝑁 ∈ ℂ) | |
5 | ax-1cn 10597 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | pncan 10894 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 588 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((𝑁 + 1) − 1) = 𝑁) |
8 | nnm1nn0 11941 | . . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) − 1) ∈ ℕ0) | |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((𝑁 + 1) − 1) ∈ ℕ0) |
10 | 7, 9 | eqeltrrd 2916 | . 2 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → 𝑁 ∈ ℕ0) |
11 | 3, 10 | impbii 211 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 1c1 10540 + caddc 10542 − cmin 10872 ℕcn 11640 ℕ0cn0 11900 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-nn 11641 df-n0 11901 |
This theorem is referenced by: elnnnn0 11943 |
Copyright terms: Public domain | W3C validator |