Theorem addnidpi 9723

Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addnidpi	⊢ (𝐴 ∈ N → ¬ (𝐴 +N 𝐵) = 𝐴)

Proof of Theorem addnidpi

Step	Hyp	Ref	Expression
1		pinn 9700	. . . . 5 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		elni2 9699	. . . . . 6 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵))
3		nnaordi 7698	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵)))
4		nna0 7684	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
5	4	eleq1d 2686	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
6		nnord 7073	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → Ord 𝐴)
7		ordirr 5741	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
8	6, 7	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
9		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +𝑜 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ 𝐴))
10	9	notbid 308	. . . . . . . . . . . . 13 ⊢ ((𝐴 +𝑜 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ ¬ 𝐴 ∈ 𝐴))
11	8, 10	syl5ibrcom 237	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
12	11	con2d 129	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
13	5, 12	sylbid 230	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
15	3, 14	syld 47	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
16	15	expcom 451	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴)))
17	16	imp32 449	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
18	2, 17	sylan2b 492	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
19	1, 18	sylan 488	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
20		addpiord 9706	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
21	20	eqeq1d 2624	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +𝑜 𝐵) = 𝐴))
22	19, 21	mtbird 315	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴)
23	22	a1d 25	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ N → ¬ (𝐴 +N 𝐵) = 𝐴))
24		dmaddpi 9712	. . . . . 6 ⊢ dom +N = (N × N)
25	24	ndmov 6818	. . . . 5 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = ∅)
26	25	eqeq1d 2624	. . . 4 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = 𝐴 ↔ ∅ = 𝐴))
27		0npi 9704	. . . . 5 ⊢ ¬ ∅ ∈ N
28		eleq1 2689	. . . . 5 ⊢ (∅ = 𝐴 → (∅ ∈ N ↔ 𝐴 ∈ N))
29	27, 28	mtbii 316	. . . 4 ⊢ (∅ = 𝐴 → ¬ 𝐴 ∈ N)
30	26, 29	syl6bi 243	. . 3 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = 𝐴 → ¬ 𝐴 ∈ N))
31	30	con2d 129	. 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ N → ¬ (𝐴 +N 𝐵) = 𝐴))
32	23, 31	pm2.61i 176	1 ⊢ (𝐴 ∈ N → ¬ (𝐴 +N 𝐵) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∅c0 3915  Ord word 5722  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557  Ncnpi 9666   +_N cpli 9667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-ni 9694  df-pli 9695

This theorem is referenced by: (None)