Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11175 | . 2 ⊢ ℕ0 ∈ V | |
2 | nnex 10903 | . 2 ⊢ ℕ ∈ V | |
3 | nn0p1nn 11209 | . 2 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 1) ∈ ℕ) | |
4 | nnm1nn0 11211 | . 2 ⊢ (𝑦 ∈ ℕ → (𝑦 − 1) ∈ ℕ0) | |
5 | nncn 10905 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
6 | nn0cn 11179 | . . 3 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
7 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | subadd 10163 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) | |
9 | 7, 8 | mp3an2 1404 | . . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) |
10 | eqcom 2617 | . . . . 5 ⊢ (𝑥 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑥) | |
11 | eqcom 2617 | . . . . 5 ⊢ (𝑦 = (1 + 𝑥) ↔ (1 + 𝑥) = 𝑦) | |
12 | 9, 10, 11 | 3bitr4g 302 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (1 + 𝑥))) |
13 | addcom 10101 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) = (𝑥 + 1)) | |
14 | 7, 13 | mpan 702 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 + 𝑥) = (𝑥 + 1)) |
15 | 14 | eqeq2d 2620 | . . . . 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 267 | . . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
18 | 5, 6, 17 | syl2anr 494 | . 2 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
19 | 1, 2, 3, 4, 18 | en3i 7880 | 1 ⊢ ℕ0 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ≈ cen 7838 ℂcc 9813 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-nn 10898 df-n0 11170 |
This theorem is referenced by: nnenom 12641 bitsf1 15006 dyadmbl 23174 aannenlem3 23889 poimirlem32 32611 heiborlem3 32782 heibor 32790 |
Copyright terms: Public domain | W3C validator |