Theorem bj-nn0sucALT 13860

Description: Alternate proof of bj-nn0suc 13846, also constructive but from ax-inf2 13858, hence requiring ax-bdsetind 13850. (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 13858	. . 3 ⊢ ∃𝑎∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧))
2		vex 2729	. . . . 5 ⊢ 𝑎 ∈ V
3		bdcv 13730	. . . . . 6 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 13856	. . . . 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 2230	. . . . . . 7 ⊢ (𝑎 = ω → (𝑦 ∈ 𝑎 ↔ 𝑦 ∈ ω))
7		rexeq 2662	. . . . . . . 8 ⊢ (𝑎 = ω → (∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
8	7	orbi2d 780	. . . . . . 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 1812	. . . . 5 ⊢ (𝑎 = ω → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
12		nfv 1516	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14		eqeq1 2172	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15		suceq 4380	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
16	15	eqeq2d 2177	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑦 = suc 𝑧 ↔ 𝑦 = suc 𝑥))
17	16	cbvrexv 2693	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18		eqeq1 2172	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
19	18	rexbidv 2467	. . . . . . . . . . . 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 783	. . . . . . . . . 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		biimp 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 2806	. . . . . . 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 4571	. . . . . . . 8 ⊢ ∅ ∈ ω
28		eleq1 2229	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
29	27, 28	mpbiri 167	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
30		bj-peano2 13821	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31		eleq1a 2238	. . . . . . . . . 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 2578	. . . . . . 7 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
35	29, 34	jaoi 706	. . . . . 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 1590	. 2 ⊢ ∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40		bj-ex 13643	. 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 698  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  Vcvv 2726  ∅c0 3409  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-nul 4108  ax-pr 4187  ax-un 4411  ax-bd0 13695  ax-bdim 13696  ax-bdor 13698  ax-bdex 13701  ax-bdeq 13702  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766  ax-bdsetind 13850  ax-inf2 13858

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568  df-bdc 13723  df-bj-ind 13809

This theorem is referenced by: (None)