Theorem bj-nn0sucALT 12146

Description: Alternate proof of bj-nn0suc 12132, also constructive but from ax-inf2 12144, hence requiring ax-bdsetind 12136. (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 12144	. . 3 ⊢ ∃𝑎∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧))
2		vex 2623	. . . . 5 ⊢ 𝑎 ∈ V
3		bdcv 12012	. . . . . 6 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 12142	. . . . 5 ⊢ (𝑎 ∈ V → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω))
5	2, 4	ax-mp 7	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω)
6		eleq2 2152	. . . . . . 7 ⊢ (𝑎 = ω → (𝑦 ∈ 𝑎 ↔ 𝑦 ∈ ω))
7		rexeq 2564	. . . . . . . 8 ⊢ (𝑎 = ω → (∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
8	7	orbi2d 740	. . . . . . 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 1753	. . . . 5 ⊢ (𝑎 = ω → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11		nfcv 2229	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
12		nfv 1467	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13		eleq1 2151	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14		eqeq1 2095	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15		suceq 4238	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
16	15	eqeq2d 2100	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑦 = suc 𝑧 ↔ 𝑦 = suc 𝑥))
17	16	cbvrexv 2592	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18		eqeq1 2095	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
19	18	rexbidv 2382	. . . . . . . . . . . 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 743	. . . . . . . . . 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 2700	. . . . . . 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 4422	. . . . . . . 8 ⊢ ∅ ∈ ω
28		eleq1 2151	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
29	27, 28	mpbiri 167	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
30		bj-peano2 12107	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31		eleq1a 2160	. . . . . . . . . 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 278	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
34	33	rexlimiva 2485	. . . . . . 7 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
35	29, 34	jaoi 672	. . . . . 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 1539	. 2 ⊢ ∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40		bj-ex 11936	. 2 ⊢ (∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
41	39, 40	ax-mp 7	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 665  ∀wal 1288   = wceq 1290  ∃wex 1427   ∈ wcel 1439  ∃wrex 2361  Vcvv 2620  ∅c0 3287  suc csuc 4201  ωcom 4418

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-nul 3971  ax-pr 4045  ax-un 4269  ax-bd0 11977  ax-bdim 11978  ax-bdor 11980  ax-bdex 11983  ax-bdeq 11984  ax-bdel 11985  ax-bdsb 11986  ax-bdsep 12048  ax-bdsetind 12136  ax-inf2 12144

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-suc 4207  df-iom 4419  df-bdc 12005  df-bj-ind 12095

This theorem is referenced by: (None)