Theorem nnsucelsuc 6702

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4612, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4634. (Contributed by Jim Kingdon, 25-Aug-2019.)

Assertion

Ref	Expression
nnsucelsuc	⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Proof of Theorem nnsucelsuc

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

Step	Hyp	Ref	Expression
1		eleq2 2295	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		suceq 4505	. . . . 5 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
3	2	eleq2d 2301	. . . 4 ⊢ (𝑥 = ∅ → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc ∅))
4	1, 3	imbi12d 234	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)))
5		eleq2 2295	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		suceq 4505	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
7	6	eleq2d 2301	. . . 4 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝑦))
8	5, 7	imbi12d 234	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦)))
9		eleq2 2295	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		suceq 4505	. . . . 5 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
11	10	eleq2d 2301	. . . 4 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc suc 𝑦))
12	9, 11	imbi12d 234	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
13		eleq2 2295	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		suceq 4505	. . . . 5 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	eleq2d 2301	. . . 4 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝐵))
16	13, 15	imbi12d 234	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵)))
17		noel 3500	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 651	. . 3 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19		elsuci 4506	. . . . . . . 8 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
20	19	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		simpl 109	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦))
22		suceq 4505	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
23	22	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦))
24	21, 23	orim12d 794	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦)))
25	20, 24	mpd 13	. . . . . 6 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
26		vex 2806	. . . . . . . 8 ⊢ 𝑦 ∈ V
27	26	sucex 4603	. . . . . . 7 ⊢ suc 𝑦 ∈ V
28	27	elsuc2 4510	. . . . . 6 ⊢ (suc 𝐴 ∈ suc suc 𝑦 ↔ (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
29	25, 28	sylibr 134	. . . . 5 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → suc 𝐴 ∈ suc suc 𝑦)
30	29	ex 115	. . . 4 ⊢ ((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) → (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦))
31	30	a1i 9	. . 3 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) → (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
32	4, 8, 12, 16, 18, 31	finds 4704	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵))
33		nnon 4714	. . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
34		onsucelsucr 4612	. . 3 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))
35	33, 34	syl 14	. 2 ⊢ (𝐵 ∈ ω → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))
36	32, 35	impbid 129	1 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2202  ∅c0 3496  Oncon0 4466  suc csuc 4468  ωcom 4694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695

This theorem is referenced by:  nnsucsssuc  6703  nntri3or  6704  nnsucuniel  6706  nnaordi  6719  ennnfonelemhom  13099