Theorem dfom2 7067

Description: An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol K_I for the inner class builder of non-limit ordinal numbers (see nlimon 7051). (Contributed by NM, 1-Nov-2004.)

Assertion

Ref	Expression
dfom2	⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Proof of Theorem dfom2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-om 7066	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)}
2		onsssuc 5813	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ suc 𝑥))
3		ontri1 5757	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
4	2, 3	bitr3d 270	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
5	4	ancoms 469	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
6		limeq 5735	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
7	6	notbid 308	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
8	7	elrab 3363	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧))
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
10	5, 9	imbi12d 334	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → ((𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
11	10	pm5.74da 723	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
12		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
13		limelon 5788	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
14	12, 13	mpan 706	. . . . . . . . . 10 ⊢ (Lim 𝑧 → 𝑧 ∈ On)
15	14	pm4.71ri 665	. . . . . . . . 9 ⊢ (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
16	15	imbi1i 339	. . . . . . . 8 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥 ∈ 𝑧))
17		impexp 462	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧 → 𝑥 ∈ 𝑧)))
18		con34b 306	. . . . . . . . . 10 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → ¬ Lim 𝑧))
19		ibar 525	. . . . . . . . . . 11 ⊢ (𝑧 ∈ On → (¬ Lim 𝑧 ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
20	19	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → ((¬ 𝑥 ∈ 𝑧 → ¬ Lim 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
21	18, 20	syl5bb 272	. . . . . . . . 9 ⊢ (𝑧 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
22	21	pm5.74i 260	. . . . . . . 8 ⊢ ((𝑧 ∈ On → (Lim 𝑧 → 𝑥 ∈ 𝑧)) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
23	16, 17, 22	3bitri 286	. . . . . . 7 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
24	11, 23	syl6rbbr 279	. . . . . 6 ⊢ (𝑥 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))))
25		impexp 462	. . . . . . 7 ⊢ (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26		simpr 477	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27		suceloni 7013	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
28		onelon 5748	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
29	28	ex 450	. . . . . . . . . . 11 ⊢ (suc 𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ On))
30	27, 29	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ On))
31	30	ancrd 577	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → (𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥)))
32	26, 31	impbid2 216	. . . . . . . 8 ⊢ (𝑥 ∈ On → ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) ↔ 𝑧 ∈ suc 𝑥))
33	32	imbi1d 331	. . . . . . 7 ⊢ (𝑥 ∈ On → (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
34	25, 33	syl5bbr 274	. . . . . 6 ⊢ (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
35	24, 34	bitrd 268	. . . . 5 ⊢ (𝑥 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
36	35	albidv 1849	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37		dfss2 3591	. . . 4 ⊢ (suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ ∀𝑧(𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
38	36, 37	syl6bbr 278	. . 3 ⊢ (𝑥 ∈ On → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
39	38	rabbiia 3185	. 2 ⊢ {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
40	1, 39	eqtri 2644	1 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ωcom 7065

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-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

This theorem is referenced by:  omsson  7069