Theorem bj-nn0sucALT 13176

Description: Alternate proof of bj-nn0suc 13162, also constructive but from ax-inf2 13174, hence requiring ax-bdsetind 13166. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-nn0sucALT	⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-nn0sucALT

Dummy variables 𝑎 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-inf2 13174	. . 3 ⊢ ∃𝑎∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧))
2		vex 2689	. . . . 5 ⊢ 𝑎 ∈ V
3		bdcv 13046	. . . . . 6 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 13172	. . . . 5 ⊢ (𝑎 ∈ V → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω))
5	2, 4	ax-mp 5	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω)
6		eleq2 2203	. . . . . . 7 ⊢ (𝑎 = ω → (𝑦 ∈ 𝑎 ↔ 𝑦 ∈ ω))
7		rexeq 2627	. . . . . . . 8 ⊢ (𝑎 = ω → (∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
8	7	orbi2d 779	. . . . . . 7 ⊢ (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
9	6, 8	bibi12d 234	. . . . . 6 ⊢ (𝑎 = ω → ((𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
10	9	albidv 1796	. . . . 5 ⊢ (𝑎 = ω → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11		nfcv 2281	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
12		nfv 1508	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13		eleq1 2202	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14		eqeq1 2146	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15		suceq 4324	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
16	15	eqeq2d 2151	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑦 = suc 𝑧 ↔ 𝑦 = suc 𝑥))
17	16	cbvrexv 2655	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18		eqeq1 2146	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
19	18	rexbidv 2438	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
20	17, 19	syl5bb 191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
21	14, 20	orbi12d 782	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
22	13, 21	bibi12d 234	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23		bi1 117	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
24	22, 23	syl6bi 162	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
25	11, 12, 24	spcimgf 2766	. . . . . . 7 ⊢ (𝐴 ∈ ω → (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
26	25	pm2.43b 52	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
27		peano1 4508	. . . . . . . 8 ⊢ ∅ ∈ ω
28		eleq1 2202	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
29	27, 28	mpbiri 167	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
30		bj-peano2 13137	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31		eleq1a 2211	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω))
32	31	imp 123	. . . . . . . . 9 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
33	30, 32	sylan 281	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
34	33	rexlimiva 2544	. . . . . . 7 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
35	29, 34	jaoi 705	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
36	26, 35	impbid1 141	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
37	10, 36	syl6bi 162	. . . 4 ⊢ (𝑎 = ω → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
38	5, 37	mpcom 36	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
39	1, 38	eximii 1581	. 2 ⊢ ∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40		bj-ex 12969	. 2 ⊢ (∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
41	39, 40	ax-mp 5	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686  ∅c0 3363  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-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdim 13012  ax-bdor 13014  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-bdsetind 13166  ax-inf2 13174

This theorem depends on definitions:  df-bi 116  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-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by: (None)