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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 11904 | . 2 ⊢ ℕ0 ∈ V | |
2 | nnex 11644 | . 2 ⊢ ℕ ∈ V | |
3 | nn0p1nn 11937 | . 2 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 1) ∈ ℕ) | |
4 | nnm1nn0 11939 | . 2 ⊢ (𝑦 ∈ ℕ → (𝑦 − 1) ∈ ℕ0) | |
5 | nncn 11646 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
6 | nn0cn 11908 | . . 3 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
7 | ax-1cn 10595 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | subadd 10889 | . . . . . 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 2828 | . . . . 5 ⊢ (𝑥 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑥) | |
11 | eqcom 2828 | . . . . 5 ⊢ (𝑦 = (1 + 𝑥) ↔ (1 + 𝑥) = 𝑦) | |
12 | 9, 10, 11 | 3bitr4g 316 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (1 + 𝑥))) |
13 | addcom 10826 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) = (𝑥 + 1)) | |
14 | 7, 13 | mpan 688 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 + 𝑥) = (𝑥 + 1)) |
15 | 14 | eqeq2d 2832 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
16 | 15 | adantl 484 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
17 | 12, 16 | bitrd 281 | . . 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 8548 | 1 ⊢ ℕ0 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ≈ cen 8506 ℂcc 10535 1c1 10538 + caddc 10540 − cmin 10870 ℕcn 11638 ℕ0cn0 11898 |
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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-n0 11899 |
This theorem is referenced by: nnenom 13349 bitsf1 15795 dyadmbl 24201 aannenlem3 24919 poimirlem32 34939 heiborlem3 35106 heibor 35114 |
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