Theorem nnsucsssuc 6460

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4486, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4504. (Contributed by Jim Kingdon, 25-Aug-2019.)

Assertion

Ref	Expression
nnsucsssuc	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem nnsucsssuc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3165	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
2		suceq 4380	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
3	2	sseq1d 3171	. . . . . 6 ⊢ (𝑥 = 𝐴 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4	1, 3	imbi12d 233	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵)))
5	4	imbi2d 229	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵)) ↔ (𝐵 ∈ ω → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))))
6		sseq1 3165	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
7		suceq 4380	. . . . . . 7 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
8	7	sseq1d 3171	. . . . . 6 ⊢ (𝑥 = ∅ → (suc 𝑥 ⊆ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
9	6, 8	imbi12d 233	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵)))
10		sseq1 3165	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
11		suceq 4380	. . . . . . 7 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
12	11	sseq1d 3171	. . . . . 6 ⊢ (𝑥 = 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc 𝑦 ⊆ suc 𝐵))
13	10, 12	imbi12d 233	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵)))
14		sseq1 3165	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝐵 ↔ suc 𝑦 ⊆ 𝐵))
15		suceq 4380	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
16	15	sseq1d 3171	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (suc 𝑥 ⊆ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
17	14, 16	imbi12d 233	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵) ↔ (suc 𝑦 ⊆ 𝐵 → suc suc 𝑦 ⊆ suc 𝐵)))
18		peano3 4573	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → suc 𝐵 ≠ ∅)
19	18	neneqd 2357	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ¬ suc 𝐵 = ∅)
20		peano2 4572	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → suc 𝐵 ∈ ω)
21		0elnn 4596	. . . . . . . . . 10 ⊢ (suc 𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
22	20, 21	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (suc 𝐵 = ∅ ∨ ∅ ∈ suc 𝐵))
23	22	ord 714	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (¬ suc 𝐵 = ∅ → ∅ ∈ suc 𝐵))
24	19, 23	mpd 13	. . . . . . 7 ⊢ (𝐵 ∈ ω → ∅ ∈ suc 𝐵)
25		nnord 4589	. . . . . . . 8 ⊢ (𝐵 ∈ ω → Ord 𝐵)
26		ordsucim 4477	. . . . . . . 8 ⊢ (Ord 𝐵 → Ord suc 𝐵)
27		0ex 4109	. . . . . . . . 9 ⊢ ∅ ∈ V
28		ordelsuc 4482	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ Ord suc 𝐵) → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
29	27, 28	mpan 421	. . . . . . . 8 ⊢ (Ord suc 𝐵 → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
30	25, 26, 29	3syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ω → (∅ ∈ suc 𝐵 ↔ suc ∅ ⊆ suc 𝐵))
31	24, 30	mpbid 146	. . . . . 6 ⊢ (𝐵 ∈ ω → suc ∅ ⊆ suc 𝐵)
32	31	a1d 22	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ ⊆ 𝐵 → suc ∅ ⊆ suc 𝐵))
33		simp3 989	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → suc 𝑦 ⊆ 𝐵)
34		simp1l 1011	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → 𝑦 ∈ ω)
35		simp1r 1012	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → 𝐵 ∈ ω)
36	35, 25	syl 14	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → Ord 𝐵)
37		ordelsuc 4482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ Ord 𝐵) → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ⊆ 𝐵))
38	34, 36, 37	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ⊆ 𝐵))
39	33, 38	mpbird 166	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → 𝑦 ∈ 𝐵)
40		nnsucelsuc 6459	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ suc 𝐵))
41	35, 40	syl 14	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ suc 𝐵))
42	39, 41	mpbid 146	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → suc 𝑦 ∈ suc 𝐵)
43		peano2 4572	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
44	34, 43	syl 14	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → suc 𝑦 ∈ ω)
45	36, 26	syl 14	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → Ord suc 𝐵)
46		ordelsuc 4482	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ ω ∧ Ord suc 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
47	44, 45, 46	syl2anc 409	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → (suc 𝑦 ∈ suc 𝐵 ↔ suc suc 𝑦 ⊆ suc 𝐵))
48	42, 47	mpbid 146	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) ∧ suc 𝑦 ⊆ 𝐵) → suc suc 𝑦 ⊆ suc 𝐵)
49	48	3expia 1195	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵)) → (suc 𝑦 ⊆ 𝐵 → suc suc 𝑦 ⊆ suc 𝐵))
50	49	exp31 362	. . . . 5 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ⊆ 𝐵 → suc 𝑦 ⊆ suc 𝐵) → (suc 𝑦 ⊆ 𝐵 → suc suc 𝑦 ⊆ suc 𝐵))))
51	9, 13, 17, 32, 50	finds2 4578	. . . 4 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ⊆ 𝐵 → suc 𝑥 ⊆ suc 𝐵)))
52	5, 51	vtoclga 2792	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵)))
53	52	imp 123	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))
54		nnon 4587	. . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
55		onsucsssucr 4486	. . 3 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
56	54, 25, 55	syl2an 287	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
57	53, 56	impbid 128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ∅c0 3409  Ord word 4340  Oncon0 4341  suc csuc 4343  ωcom 4567

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568

This theorem is referenced by:  nnaword  6479  ennnfonelemk  12333  ennnfonelemkh  12345