Theorem nnsucelsuc 6635

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4599, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4621. (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 2293	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		suceq 4492	. . . . 5 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
3	2	eleq2d 2299	. . . 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 2293	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		suceq 4492	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
7	6	eleq2d 2299	. . . 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 2293	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		suceq 4492	. . . . 5 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
11	10	eleq2d 2299	. . . 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 2293	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		suceq 4492	. . . . 5 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	eleq2d 2299	. . . 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 3495	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 649	. . 3 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19		elsuci 4493	. . . . . . . 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 4492	. . . . . . . . 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 791	. . . . . . 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 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
27	26	sucex 4590	. . . . . . 7 ⊢ suc 𝑦 ∈ V
28	27	elsuc2 4497	. . . . . 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 4691	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵))
33		nnon 4701	. . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
34		onsucelsucr 4599	. . 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 713   = wceq 1395   ∈ wcel 2200  ∅c0 3491  Oncon0 4453  suc csuc 4455  ωcom 4681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682

This theorem is referenced by:  nnsucsssuc  6636  nntri3or  6637  nnsucuniel  6639  nnaordi  6652  ennnfonelemhom  12981