Theorem nnsucelsuc 6470

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4492, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4514. (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 2234	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		suceq 4387	. . . . 5 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
3	2	eleq2d 2240	. . . 4 ⊢ (𝑥 = ∅ → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc ∅))
4	1, 3	imbi12d 233	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)))
5		eleq2 2234	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		suceq 4387	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
7	6	eleq2d 2240	. . . 4 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝑦))
8	5, 7	imbi12d 233	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦)))
9		eleq2 2234	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		suceq 4387	. . . . 5 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
11	10	eleq2d 2240	. . . 4 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc suc 𝑦))
12	9, 11	imbi12d 233	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ suc 𝑦 → suc 𝐴 ∈ suc suc 𝑦)))
13		eleq2 2234	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		suceq 4387	. . . . 5 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	eleq2d 2240	. . . 4 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ∈ suc 𝑥 ↔ suc 𝐴 ∈ suc 𝐵))
16	13, 15	imbi12d 233	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → suc 𝐴 ∈ suc 𝑥) ↔ (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵)))
17		noel 3418	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 641	. . 3 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19		elsuci 4388	. . . . . . . 8 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
20	19	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		simpl 108	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → (𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦))
22		suceq 4387	. . . . . . . . 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 781	. . . . . . 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 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
27	26	sucex 4483	. . . . . . 7 ⊢ suc 𝑦 ∈ V
28	27	elsuc2 4392	. . . . . 6 ⊢ (suc 𝐴 ∈ suc suc 𝑦 ↔ (suc 𝐴 ∈ suc 𝑦 ∨ suc 𝐴 = suc 𝑦))
29	25, 28	sylibr 133	. . . . 5 ⊢ (((𝐴 ∈ 𝑦 → suc 𝐴 ∈ suc 𝑦) ∧ 𝐴 ∈ suc 𝑦) → suc 𝐴 ∈ suc suc 𝑦)
30	29	ex 114	. . . 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 4584	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵))
33		nnon 4594	. . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
34		onsucelsucr 4492	. . 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 128	1 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   = wceq 1348   ∈ wcel 2141  ∅c0 3414  Oncon0 4348  suc csuc 4350  ωcom 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575

This theorem is referenced by:  nnsucsssuc  6471  nntri3or  6472  nnsucuniel  6474  nnaordi  6487  ennnfonelemhom  12370