Theorem bj-nn0sucALT 16740

Description: Alternate proof of bj-nn0suc 16726, also constructive but from ax-inf2 16738, hence requiring ax-bdsetind 16730. (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 16738	. . 3 ⊢ ∃𝑎∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧))
2		vex 2815	. . . . 5 ⊢ 𝑎 ∈ V
3		bdcv 16610	. . . . . 6 ⊢ BOUNDED 𝑎
4	3	bj-inf2vn 16736	. . . . 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 2296	. . . . . . 7 ⊢ (𝑎 = ω → (𝑦 ∈ 𝑎 ↔ 𝑦 ∈ ω))
7		rexeq 2741	. . . . . . . 8 ⊢ (𝑎 = ω → (∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
8	7	orbi2d 798	. . . . . . 7 ⊢ (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
9	6, 8	bibi12d 235	. . . . . 6 ⊢ (𝑎 = ω → ((𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
10	9	albidv 1873	. . . . 5 ⊢ (𝑎 = ω → (∀𝑦(𝑦 ∈ 𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ 𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
12		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13		eleq1 2295	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14		eqeq1 2239	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15		suceq 4522	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
16	15	eqeq2d 2244	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑦 = suc 𝑧 ↔ 𝑦 = suc 𝑥))
17	16	cbvrexv 2778	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18		eqeq1 2239	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
19	18	rexbidv 2543	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
20	17, 19	bitrid 192	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
21	14, 20	orbi12d 801	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
22	13, 21	bibi12d 235	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23		biimp 118	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
24	22, 23	biimtrdi 163	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
25	11, 12, 24	spcimgf 2896	. . . . . . 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 4715	. . . . . . . 8 ⊢ ∅ ∈ ω
28		eleq1 2295	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
29	27, 28	mpbiri 168	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
30		bj-peano2 16701	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31		eleq1a 2304	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω))
32	31	imp 124	. . . . . . . . 9 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
33	30, 32	sylan 283	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
34	33	rexlimiva 2655	. . . . . . 7 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
35	29, 34	jaoi 724	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
36	26, 35	impbid1 142	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
37	10, 36	biimtrdi 163	. . . 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 1651	. 2 ⊢ ∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40		bj-ex 16526	. 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 105   ∨ wo 716  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  Vcvv 2812  ∅c0 3507  suc csuc 4485  ωcom 4711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4235  ax-pr 4321  ax-un 4553  ax-bd0 16575  ax-bdim 16576  ax-bdor 16578  ax-bdex 16581  ax-bdeq 16582  ax-bdel 16583  ax-bdsb 16584  ax-bdsep 16646  ax-bdsetind 16730  ax-inf2 16738

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-sn 3694  df-pr 3695  df-uni 3914  df-int 3949  df-suc 4491  df-iom 4712  df-bdc 16603  df-bj-ind 16689

This theorem is referenced by: (None)