Theorem bnj168 30798

Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8497. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj168.1	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj168	⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚)

Distinct variable group:   𝑚,𝑛

Allowed substitution hints:   𝐷(𝑚,𝑛)

Proof of Theorem bnj168

Step	Hyp	Ref	Expression
1		bnj168.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj158 30797	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
3	2	anim2i 593	. . . . . . . 8 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4		r19.42v 3092	. . . . . . . 8 ⊢ (∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
5	3, 4	sylibr 224	. . . . . . 7 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚))
6		neeq1 2856	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑚 → (𝑛 ≠ 1𝑜 ↔ suc 𝑚 ≠ 1𝑜))
7	6	biimpac 503	. . . . . . . . . 10 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) → suc 𝑚 ≠ 1𝑜)
8		df-1o 7560	. . . . . . . . . . . . 13 ⊢ 1𝑜 = suc ∅
9	8	eqeq2i 2634	. . . . . . . . . . . 12 ⊢ (suc 𝑚 = 1𝑜 ↔ suc 𝑚 = suc ∅)
10		nnon 7071	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
11		0elon 5778	. . . . . . . . . . . . 13 ⊢ ∅ ∈ On
12		suc11 5831	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
13	10, 11, 12	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
14	9, 13	syl5rbb 273	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1𝑜))
15	14	necon3bid 2838	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1𝑜))
16	7, 15	syl5ibr 236	. . . . . . . . 9 ⊢ (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
17	16	ancld 576	. . . . . . . 8 ⊢ (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) → ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
18	17	reximia 3009	. . . . . . 7 ⊢ (∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) → ∃𝑚 ∈ ω ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
19	5, 18	syl 17	. . . . . 6 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ ω ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
20		anass 681	. . . . . . 7 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)))
21	20	rexbii 3041	. . . . . 6 ⊢ (∃𝑚 ∈ ω ((𝑛 ≠ 1𝑜 ∧ 𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)))
22	19, 21	sylib 208	. . . . 5 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)))
23		simpr 477	. . . . 5 ⊢ ((𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅))
24	22, 23	bnj31 30785	. . . 4 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅))
25		df-rex 2918	. . . 4 ⊢ (∃𝑚 ∈ ω (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)))
26	24, 25	sylib 208	. . 3 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)))
27		simpr 477	. . . . . . 7 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅) → 𝑚 ≠ ∅)
28	27	anim2i 593	. . . . . 6 ⊢ ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
29	1	eleq2i 2693	. . . . . . 7 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
30		eldifsn 4317	. . . . . . 7 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
31	29, 30	bitr2i 265	. . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚 ∈ 𝐷)
32	28, 31	sylib 208	. . . . 5 ⊢ ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → 𝑚 ∈ 𝐷)
33		simprl 794	. . . . 5 ⊢ ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
34	32, 33	jca 554	. . . 4 ⊢ ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
35	34	eximi 1762	. . 3 ⊢ (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚 ∧ 𝑚 ≠ ∅)) → ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
36	26, 35	syl 17	. 2 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
37		df-rex 2918	. 2 ⊢ (∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
38	36, 37	sylibr 224	1 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571  ∅c0 3915  {csn 4177  Oncon0 5723  suc csuc 5725  ωcom 7065  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-1o 7560

This theorem is referenced by:  bnj600  30989