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Mirrors > Home > ILE Home > Th. List > addnidpig | GIF version |
Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
Ref | Expression |
---|---|
addnidpig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6407 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | elni2 6412 | . . . 4 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
3 | nnaordi 6081 | . . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) | |
4 | nna0 6053 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴) | |
5 | 4 | eleq1d 2106 | . . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ (𝐴 +𝑜 𝐵))) |
6 | nnord 4334 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | ordirr 4267 | . . . . . . . . . . . 12 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | 6, 7 | syl 14 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) |
9 | eleq2 2101 | . . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ 𝐴)) | |
10 | 9 | notbid 592 | . . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ ¬ 𝐴 ∈ 𝐴)) |
11 | 8, 10 | syl5ibrcom 146 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +𝑜 𝐵))) |
12 | 11 | con2d 554 | . . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)) |
13 | 5, 12 | sylbid 139 | . . . . . . . 8 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)) |
14 | 13 | adantl 262 | . . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)) |
15 | 3, 14 | syld 40 | . . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴)) |
16 | 15 | expcom 109 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴))) |
17 | 16 | imp32 244 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = 𝐴) |
18 | 2, 17 | sylan2b 271 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴) |
19 | 1, 18 | sylan 267 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴) |
20 | addpiord 6414 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵)) | |
21 | 20 | eqeq1d 2048 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +𝑜 𝐵) = 𝐴)) |
22 | 19, 21 | mtbird 598 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∅c0 3224 Ord word 4099 ωcom 4313 (class class class)co 5512 +𝑜 coa 5998 Ncnpi 6370 +N cpli 6371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-ni 6402 df-pli 6403 |
This theorem is referenced by: (None) |
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