Theorem nnn1suc 41482

Description: A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)

Assertion

Ref	Expression
nnn1suc	⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem nnn1suc

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 3001	. . . 4 ⊢ (𝑦 = 1 → (𝑦 ≠ 1 ↔ 1 ≠ 1))
2		eqeq2 2742	. . . . 5 ⊢ (𝑦 = 1 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 1))
3	2	rexbidv 3176	. . . 4 ⊢ (𝑦 = 1 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 1))
4	1, 3	imbi12d 343	. . 3 ⊢ (𝑦 = 1 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (1 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)))
5		neeq1 3001	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ≠ 1 ↔ 𝑧 ≠ 1))
6		eqeq2 2742	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 𝑧))
7	6	rexbidv 3176	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧))
8	5, 7	imbi12d 343	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (𝑧 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧)))
9		neeq1 3001	. . . 4 ⊢ (𝑦 = (𝑧 + 1) → (𝑦 ≠ 1 ↔ (𝑧 + 1) ≠ 1))
10		eqeq2 2742	. . . . 5 ⊢ (𝑦 = (𝑧 + 1) → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = (𝑧 + 1)))
11	10	rexbidv 3176	. . . 4 ⊢ (𝑦 = (𝑧 + 1) → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1)))
12	9, 11	imbi12d 343	. . 3 ⊢ (𝑦 = (𝑧 + 1) → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ ((𝑧 + 1) ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))))
13		neeq1 3001	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ 1 ↔ 𝐴 ≠ 1))
14		eqeq2 2742	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 𝐴))
15	14	rexbidv 3176	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴))
16	13, 15	imbi12d 343	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (𝐴 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)))
17		df-ne 2939	. . . 4 ⊢ (1 ≠ 1 ↔ ¬ 1 = 1)
18		eqid 2730	. . . . 5 ⊢ 1 = 1
19	18	pm2.24i 150	. . . 4 ⊢ (¬ 1 = 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)
20	17, 19	sylbi 216	. . 3 ⊢ (1 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)
21		id 22	. . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
22		oveq1 7418	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
23	22	eqeq1d 2732	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 + 1) = (𝑧 + 1) ↔ (𝑧 + 1) = (𝑧 + 1)))
24	23	adantl 480	. . . . 5 ⊢ ((𝑧 ∈ ℕ ∧ 𝑥 = 𝑧) → ((𝑥 + 1) = (𝑧 + 1) ↔ (𝑧 + 1) = (𝑧 + 1)))
25		eqidd 2731	. . . . 5 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) = (𝑧 + 1))
26	21, 24, 25	rspcedvd 3613	. . . 4 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))
27	26	2a1d 26	. . 3 ⊢ (𝑧 ∈ ℕ → ((𝑧 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧) → ((𝑧 + 1) ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))))
28	4, 8, 12, 16, 20, 27	nnind 12234	. 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴))
29	28	imp 405	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∃wrex 3068  (class class class)co 7411  1c1 11113   + caddc 11115  ℕcn 12216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727  ax-1cn 11170

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217

This theorem is referenced by: (None)