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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 5714 | . . 3 ⊢ (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅)) | |
2 | id 19 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
3 | 1, 2 | eqeq12d 2114 | . 2 ⊢ (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅)) |
4 | oveq2 5714 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦)) | |
5 | id 19 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
6 | 4, 5 | eqeq12d 2114 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦)) |
7 | oveq2 5714 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦)) | |
8 | id 19 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
9 | 7, 8 | eqeq12d 2114 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦)) |
10 | oveq2 5714 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴)) | |
11 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
12 | 10, 11 | eqeq12d 2114 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴)) |
13 | 0elon 4252 | . . 3 ⊢ ∅ ∈ On | |
14 | oa0 6283 | . . 3 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
15 | 13, 14 | ax-mp 7 | . 2 ⊢ (∅ +o ∅) = ∅ |
16 | peano1 4446 | . . . 4 ⊢ ∅ ∈ ω | |
17 | nnasuc 6302 | . . . 4 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) | |
18 | 16, 17 | mpan 418 | . . 3 ⊢ (𝑦 ∈ ω → (∅ +o suc 𝑦) = suc (∅ +o 𝑦)) |
19 | suceq 4262 | . . . 4 ⊢ ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦) | |
20 | 19 | eqeq2d 2111 | . . 3 ⊢ ((∅ +o 𝑦) = 𝑦 → ((∅ +o suc 𝑦) = suc (∅ +o 𝑦) ↔ (∅ +o suc 𝑦) = suc 𝑦)) |
21 | 18, 20 | syl5ibcom 154 | . 2 ⊢ (𝑦 ∈ ω → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦)) |
22 | 3, 6, 9, 12, 15, 21 | finds 4452 | 1 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∈ wcel 1448 ∅c0 3310 Oncon0 4223 suc csuc 4225 ωcom 4442 (class class class)co 5706 +o coa 6240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-oadd 6247 |
This theorem is referenced by: nnacom 6310 nnaword 6337 nnm1 6350 prarloclem5 7209 |
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