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Mirrors > Home > ILE Home > Th. List > nna0r | GIF version |
Description: Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
nna0r | ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5573 | . . 3 ⊢ (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅)) | |
2 | id 19 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2097 | . 2 ⊢ (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅)) |
4 | oveq2 5573 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦)) | |
5 | id 19 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2097 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦)) |
7 | oveq2 5573 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦)) | |
8 | id 19 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2097 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
10 | oveq2 5573 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴)) | |
11 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2097 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴)) |
13 | 0elon 4176 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 6123 | . . 3 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
15 | 13, 14 | ax-mp 7 | . 2 ⊢ (∅ +𝑜 ∅) = ∅ |
16 | peano1 4364 | . . . 4 ⊢ ∅ ∈ ω | |
17 | nnasuc 6142 | . . . 4 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) | |
18 | 16, 17 | mpan 415 | . . 3 ⊢ (𝑦 ∈ ω → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦)) |
19 | suceq 4186 | . . . 4 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦) | |
20 | 19 | eqeq2d 2094 | . . 3 ⊢ ((∅ +𝑜 𝑦) = 𝑦 → ((∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦) ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
21 | 18, 20 | syl5ibcom 153 | . 2 ⊢ (𝑦 ∈ ω → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦)) |
22 | 3, 6, 9, 12, 15, 21 | finds 4370 | 1 ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ∅c0 3268 Oncon0 4147 suc csuc 4149 ωcom 4360 (class class class)co 5565 +𝑜 coa 6084 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4077 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-irdg 6041 df-oadd 6091 |
This theorem is referenced by: nnacom 6150 nnaword 6173 nnm1 6186 prarloclem5 6829 |
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