Theorem addnidpi 3822

Description: There is no identity element for addition on positive integers.

Hypothesis

Ref	Expression
addnidpi.1	⊢ B ∈ V

Assertion

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

Proof of Theorem addnidpi

Step	Hyp	Ref	Expression
1		nnaordi 3176	. . . . . . . . . 10 ⊢ ((B ∈ ω ∧ A ∈ ω) → (∅ ∈ B → (A +o ∅) ∈ (A +o B)))
2		nna0 3166	. . . . . . . . . . . . 13 ⊢ (A ∈ ω → (A +o ∅) = A)
3	2	eleq1d 1155	. . . . . . . . . . . 12 ⊢ (A ∈ ω → ((A +o ∅) ∈ (A +o B) ↔ A ∈ (A +o B)))
4		eleq2 1150	. . . . . . . . . . . . . . . 16 ⊢ ((A +o B) = A → (A ∈ (A +o B) ↔ A ∈ A))
5	4	negbid 463	. . . . . . . . . . . . . . 15 ⊢ ((A +o B) = A → (¬ A ∈ (A +o B) ↔ ¬ A ∈ A))
6		nnord 2381	. . . . . . . . . . . . . . . 16 ⊢ (A ∈ ω → Ord A)
7		ordeirr 2217	. . . . . . . . . . . . . . . 16 ⊢ (Ord A → ¬ A ∈ A)
8	6, 7	syl 12	. . . . . . . . . . . . . . 15 ⊢ (A ∈ ω → ¬ A ∈ A)
9	5, 8	syl5bir 184	. . . . . . . . . . . . . 14 ⊢ ((A +o B) = A → (A ∈ ω → ¬ A ∈ (A +o B)))
10	9	com12 13	. . . . . . . . . . . . 13 ⊢ (A ∈ ω → ((A +o B) = A → ¬ A ∈ (A +o B)))
11	10	con2d 83	. . . . . . . . . . . 12 ⊢ (A ∈ ω → (A ∈ (A +o B) → ¬ (A +o B) = A))
12	3, 11	sylbid 178	. . . . . . . . . . 11 ⊢ (A ∈ ω → ((A +o ∅) ∈ (A +o B) → ¬ (A +o B) = A))
13	12	adantl 305	. . . . . . . . . 10 ⊢ ((B ∈ ω ∧ A ∈ ω) → ((A +o ∅) ∈ (A +o B) → ¬ (A +o B) = A))
14	1, 13	syld 27	. . . . . . . . 9 ⊢ ((B ∈ ω ∧ A ∈ ω) → (∅ ∈ B → ¬ (A +o B) = A))
15	14	exp 291	. . . . . . . 8 ⊢ (B ∈ ω → (A ∈ ω → (∅ ∈ B → ¬ (A +o B) = A)))
16	15	com12 13	. . . . . . 7 ⊢ (A ∈ ω → (B ∈ ω → (∅ ∈ B → ¬ (A +o B) = A)))
17	16	imp32 281	. . . . . 6 ⊢ ((A ∈ ω ∧ (B ∈ ω ∧ ∅ ∈ B)) → ¬ (A +o B) = A)
18		elni2 3799	. . . . . 6 ⊢ (B ∈ N ↔ (B ∈ ω ∧ ∅ ∈ B))
19	17, 18	sylan2b 347	. . . . 5 ⊢ ((A ∈ ω ∧ B ∈ N) → ¬ (A +o B) = A)
20		pinn 3800	. . . . 5 ⊢ (A ∈ N → A ∈ ω)
21	19, 20	sylan 343	. . . 4 ⊢ ((A ∈ N ∧ B ∈ N) → ¬ (A +o B) = A)
22		addpiord 3806	. . . . 5 ⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) = (A +o B))
23	22	cleq1d 1109	. . . 4 ⊢ ((A ∈ N ∧ B ∈ N) → ((A +N B) = A ↔ (A +o B) = A))
24	21, 23	mtbird 537	. . 3 ⊢ ((A ∈ N ∧ B ∈ N) → ¬ (A +N B) = A)
25	24	a1d 14	. 2 ⊢ ((A ∈ N ∧ B ∈ N) → (A ∈ N → ¬ (A +N B) = A))
26		addnidpi.1	. . . . . 6 ⊢ B ∈ V
27		dmaddpi 3812	. . . . . 6 ⊢ dom +N = (N × N)
28	26, 27	ndmopr 3059	. . . . 5 ⊢ (¬ (A ∈ N ∧ B ∈ N) → (A +N B) = ∅)
29	28	cleq1d 1109	. . . 4 ⊢ (¬ (A ∈ N ∧ B ∈ N) → ((A +N B) = A ↔ ∅ = A))
30		0npi 3804	. . . . 5 ⊢ ¬ ∅ ∈ N
31		eleq1 1149	. . . . 5 ⊢ (∅ = A → (∅ ∈ N ↔ A ∈ N))
32	30, 31	mtbii 538	. . . 4 ⊢ (∅ = A → ¬ A ∈ N)
33	29, 32	syl6bi 187	. . 3 ⊢ (¬ (A ∈ N ∧ B ∈ N) → ((A +N B) = A → ¬ A ∈ N))
34	33	con2d 83	. 2 ⊢ (¬ (A ∈ N ∧ B ∈ N) → (A ∈ N → ¬ (A +N B) = A))
35	25, 34	pm2.61i 110	1 ⊢ (A ∈ N → ¬ (A +N B) = A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  Ord word 2198  ωcom 2372  (class class class)co 3001   +_o coa 3101  Ncnpi 3766   +_N cpli 3767

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-ni 3794  df-pli 3795

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