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Mirrors > Home > MPE Home > Th. List > nn0ennn | Structured version Visualization version GIF version |
Description: The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
Ref | Expression |
---|---|
nn0ennn | ⊢ ℕ0 ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ex 12477 | . 2 ⊢ ℕ0 ∈ V | |
2 | nnex 12217 | . 2 ⊢ ℕ ∈ V | |
3 | nn0p1nn 12510 | . 2 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 1) ∈ ℕ) | |
4 | nnm1nn0 12512 | . 2 ⊢ (𝑦 ∈ ℕ → (𝑦 − 1) ∈ ℕ0) | |
5 | nncn 12219 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
6 | nn0cn 12481 | . . 3 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
7 | ax-1cn 11165 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | subadd 11462 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) | |
9 | 7, 8 | mp3an2 1445 | . . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) |
10 | eqcom 2731 | . . . . 5 ⊢ (𝑥 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑥) | |
11 | eqcom 2731 | . . . . 5 ⊢ (𝑦 = (1 + 𝑥) ↔ (1 + 𝑥) = 𝑦) | |
12 | 9, 10, 11 | 3bitr4g 314 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (1 + 𝑥))) |
13 | addcom 11399 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) = (𝑥 + 1)) | |
14 | 7, 13 | mpan 687 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 + 𝑥) = (𝑥 + 1)) |
15 | 14 | eqeq2d 2735 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
16 | 15 | adantl 481 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
17 | 12, 16 | bitrd 279 | . . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
18 | 5, 6, 17 | syl2anr 596 | . 2 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
19 | 1, 2, 3, 4, 18 | en3i 8984 | 1 ⊢ ℕ0 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5139 (class class class)co 7402 ≈ cen 8933 ℂcc 11105 1c1 11108 + caddc 11110 − cmin 11443 ℕcn 12211 ℕ0cn0 12471 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-ltxr 11252 df-sub 11445 df-nn 12212 df-n0 12472 |
This theorem is referenced by: nnenom 13946 bitsf1 16390 dyadmbl 25473 aannenlem3 26208 poimirlem32 37024 heiborlem3 37185 heibor 37193 |
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