Theorem nnsucelsuc 6387

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4424, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4445. (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 2203	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		suceq 4324	. . . . 5 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
3	2	eleq2d 2209	. . . 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 2203	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		suceq 4324	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
7	6	eleq2d 2209	. . . 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 2203	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		suceq 4324	. . . . 5 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
11	10	eleq2d 2209	. . . 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 2203	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		suceq 4324	. . . . 5 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
15	14	eleq2d 2209	. . . 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 3367	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 635	. . 3 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ suc ∅)
19		elsuci 4325	. . . . . . . 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 4324	. . . . . . . . 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 775	. . . . . . 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 2689	. . . . . . . 8 ⊢ 𝑦 ∈ V
27	26	sucex 4415	. . . . . . 7 ⊢ suc 𝑦 ∈ V
28	27	elsuc2 4329	. . . . . 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 4514	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ suc 𝐵))
33		nnon 4523	. . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
34		onsucelsucr 4424	. . 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 697   = wceq 1331   ∈ wcel 1480  ∅c0 3363  Oncon0 4285  suc csuc 4287  ωcom 4504

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505

This theorem is referenced by:  nnsucsssuc  6388  nntri3or  6389  nnsucuniel  6391  nnaordi  6404  ennnfonelemhom  11931