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Mirrors > Home > ILE Home > Th. List > nn0ennn | 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 8951 | . 2 ⊢ ℕ0 ∈ V | |
2 | nnex 8694 | . 2 ⊢ ℕ ∈ V | |
3 | nn0p1nn 8984 | . 2 ⊢ (𝑥 ∈ ℕ0 → (𝑥 + 1) ∈ ℕ) | |
4 | nnm1nn0 8986 | . 2 ⊢ (𝑦 ∈ ℕ → (𝑦 − 1) ∈ ℕ0) | |
5 | nncn 8696 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
6 | nn0cn 8955 | . . 3 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ) | |
7 | ax-1cn 7681 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | subadd 7933 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) | |
9 | 7, 8 | mp3an2 1288 | . . . . 5 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 − 1) = 𝑥 ↔ (1 + 𝑥) = 𝑦)) |
10 | eqcom 2119 | . . . . 5 ⊢ (𝑥 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑥) | |
11 | eqcom 2119 | . . . . 5 ⊢ (𝑦 = (1 + 𝑥) ↔ (1 + 𝑥) = 𝑦) | |
12 | 9, 10, 11 | 3bitr4g 222 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (1 + 𝑥))) |
13 | addcom 7867 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) = (𝑥 + 1)) | |
14 | 7, 13 | mpan 420 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 + 𝑥) = (𝑥 + 1)) |
15 | 14 | eqeq2d 2129 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
16 | 15 | adantl 275 | . . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 = (1 + 𝑥) ↔ 𝑦 = (𝑥 + 1))) |
17 | 12, 16 | bitrd 187 | . . 3 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
18 | 5, 6, 17 | syl2anr 288 | . 2 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 − 1) ↔ 𝑦 = (𝑥 + 1))) |
19 | 1, 2, 3, 4, 18 | en3i 6633 | 1 ⊢ ℕ0 ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ≈ cen 6600 ℂcc 7586 1c1 7589 + caddc 7591 − cmin 7901 ℕcn 8688 ℕ0cn0 8945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-en 6603 df-sub 7903 df-inn 8689 df-n0 8946 |
This theorem is referenced by: nnenom 10175 uzennn 10177 xpnnen 11834 znnen 11838 ennnfonelemim 11864 |
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