Theorem nnsucelsuc 6002

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4198, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4214. (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 2098	. . . 4 ⊢ (x = ∅ → (A ∈ x ↔ A ∈ ∅))
2		suceq 4104	. . . . 5 ⊢ (x = ∅ → suc x = suc ∅)
3	2	eleq2d 2104	. . . 4 ⊢ (x = ∅ → (suc A ∈ suc x ↔ suc A ∈ suc ∅))
4	1, 3	imbi12d 223	. . 3 ⊢ (x = ∅ → ((A ∈ x → suc A ∈ suc x) ↔ (A ∈ ∅ → suc A ∈ suc ∅)))
5		eleq2 2098	. . . 4 ⊢ (x = y → (A ∈ x ↔ A ∈ y))
6		suceq 4104	. . . . 5 ⊢ (x = y → suc x = suc y)
7	6	eleq2d 2104	. . . 4 ⊢ (x = y → (suc A ∈ suc x ↔ suc A ∈ suc y))
8	5, 7	imbi12d 223	. . 3 ⊢ (x = y → ((A ∈ x → suc A ∈ suc x) ↔ (A ∈ y → suc A ∈ suc y)))
9		eleq2 2098	. . . 4 ⊢ (x = suc y → (A ∈ x ↔ A ∈ suc y))
10		suceq 4104	. . . . 5 ⊢ (x = suc y → suc x = suc suc y)
11	10	eleq2d 2104	. . . 4 ⊢ (x = suc y → (suc A ∈ suc x ↔ suc A ∈ suc suc y))
12	9, 11	imbi12d 223	. . 3 ⊢ (x = suc y → ((A ∈ x → suc A ∈ suc x) ↔ (A ∈ suc y → suc A ∈ suc suc y)))
13		eleq2 2098	. . . 4 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
14		suceq 4104	. . . . 5 ⊢ (x = B → suc x = suc B)
15	14	eleq2d 2104	. . . 4 ⊢ (x = B → (suc A ∈ suc x ↔ suc A ∈ suc B))
16	13, 15	imbi12d 223	. . 3 ⊢ (x = B → ((A ∈ x → suc A ∈ suc x) ↔ (A ∈ B → suc A ∈ suc B)))
17		noel 3222	. . . 4 ⊢ ¬ A ∈ ∅
18	17	pm2.21i 574	. . 3 ⊢ (A ∈ ∅ → suc A ∈ suc ∅)
19		elsuci 4105	. . . . . . . 8 ⊢ (A ∈ suc y → (A ∈ y ∨ A = y))
20	19	adantl 262	. . . . . . 7 ⊢ (((A ∈ y → suc A ∈ suc y) ∧ A ∈ suc y) → (A ∈ y ∨ A = y))
21		simpl 102	. . . . . . . 8 ⊢ (((A ∈ y → suc A ∈ suc y) ∧ A ∈ suc y) → (A ∈ y → suc A ∈ suc y))
22		suceq 4104	. . . . . . . . 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 699	. . . . . . 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 2554	. . . . . . . 8 ⊢ y ∈ V
27	26	sucex 4190	. . . . . . 7 ⊢ suc y ∈ V
28	27	elsuc2 4109	. . . . . 6 ⊢ (suc A ∈ suc suc y ↔ (suc A ∈ suc y ∨ suc A = suc y))
29	25, 28	sylibr 137	. . . . 5 ⊢ (((A ∈ y → suc A ∈ suc y) ∧ A ∈ suc y) → suc A ∈ suc suc y)
30	29	ex 108	. . . 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 4265	. 2 ⊢ (B ∈ 𝜔 → (A ∈ B → suc A ∈ suc B))
33		nnon 4274	. . 3 ⊢ (B ∈ 𝜔 → B ∈ On)
34		onsucelsucr 4198	. . 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 120	1 ⊢ (B ∈ 𝜔 → (A ∈ B ↔ suc A ∈ suc B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∅c0 3218  Oncon0 4065  suc csuc 4067  𝜔com 4255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-iinf 4253

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-int 3606  df-tr 3845  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256

This theorem is referenced by:  nnsucsssuc  6003  nntri3or  6004  nnaordi  6010