Theorem nnsucelsuc 6093

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4252, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4273. (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 2142	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		suceq 4157	. . . . 5 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
3	2	eleq2d 2148	. . . 4 ⊢ (𝑥 = ∅ → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc ∅))
4	1, 3	imbi12d 232	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)))
5		eleq2 2142	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		suceq 4157	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
7	6	eleq2d 2148	. . . 4 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝑦))
8	5, 7	imbi12d 232	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦)))
9		eleq2 2142	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		suceq 4157	. . . . 5 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
11	10	eleq2d 2148	. . . 4 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc suc 𝑦))
12	9, 11	imbi12d 232	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
13		eleq2 2142	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		suceq 4157	. . . . 5 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	eleq2d 2148	. . . 4 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝐵))
16	13, 15	imbi12d 232	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵)))
17		noel 3255	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 607	. . 3 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19		elsuci 4158	. . . . . . . 8 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
20	19	adantl 271	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		simpl 107	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦))
22		suceq 4157	. . . . . . . . 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 732	. . . . . . 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 2604	. . . . . . . 8 ⊢ 𝑦 ∈ V
27	26	sucex 4243	. . . . . . 7 ⊢ suc 𝑦 ∈ V
28	27	elsuc2 4162	. . . . . 6 ⊢ (suc 𝐴 ∈ suc suc 𝑦 ↔ (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
29	25, 28	sylibr 132	. . . . 5 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → suc 𝐴 ∈ suc suc 𝑦)
30	29	ex 113	. . . 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 4341	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵))
33		nnon 4350	. . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
34		onsucelsucr 4252	. . 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 127	1 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   = wceq 1284   ∈ wcel 1433  ∅c0 3251  Oncon0 4118  suc csuc 4120  ωcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332

This theorem is referenced by:  nnsucsssuc  6094  nntri3or  6095  nnaordi  6104