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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 12427 | . 2 ⊢ ℕ0 ∈ V | |
2 | nnex 12167 | . 2 ⊢ ℕ ∈ V | |
3 | nn0p1nn 12460 | . 2 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 1) ∈ ℕ) | |
4 | nnm1nn0 12462 | . 2 ⊢ (𝑦 ∈ ℕ → (𝑦 − 1) ∈ ℕ0) | |
5 | nncn 12169 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
6 | nn0cn 12431 | . . 3 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
7 | ax-1cn 11117 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | subadd 11412 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) | |
9 | 7, 8 | mp3an2 1450 | . . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) |
10 | eqcom 2740 | . . . . 5 ⊢ (𝑥 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑥) | |
11 | eqcom 2740 | . . . . 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 11349 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) = (𝑥 + 1)) | |
14 | 7, 13 | mpan 689 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 + 𝑥) = (𝑥 + 1)) |
15 | 14 | eqeq2d 2744 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
16 | 15 | adantl 483 | . . . 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 598 | . 2 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
19 | 1, 2, 3, 4, 18 | en3i 8937 | 1 ⊢ ℕ0 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5109 (class class class)co 7361 ≈ cen 8886 ℂcc 11057 1c1 11060 + caddc 11062 − cmin 11393 ℕcn 12161 ℕ0cn0 12421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-sub 11395 df-nn 12162 df-n0 12422 |
This theorem is referenced by: nnenom 13894 bitsf1 16334 dyadmbl 24987 aannenlem3 25713 poimirlem32 36160 heiborlem3 36322 heibor 36330 |
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