Theorem dfom2 7810

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 restricted class abstraction of non-limit ordinal numbers (see nlimon 7793). (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 7809	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)}
2		vex 3444	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
3		limelon 6382	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
4	2, 3	mpan 690	. . . . . . . . . 10 ⊢ (Lim 𝑧 → 𝑧 ∈ On)
5	4	pm4.71ri 560	. . . . . . . . 9 ⊢ (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
6	5	imbi1i 349	. . . . . . . 8 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥 ∈ 𝑧))
7		impexp 450	. . . . . . . 8 ⊢ (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧 → 𝑥 ∈ 𝑧)))
8		con34b 316	. . . . . . . . . 10 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → ¬ Lim 𝑧))
9		ibar 528	. . . . . . . . . . 11 ⊢ (𝑧 ∈ On → (¬ Lim 𝑧 ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
10	9	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → ((¬ 𝑥 ∈ 𝑧 → ¬ Lim 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
11	8, 10	bitrid 283	. . . . . . . . 9 ⊢ (𝑧 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
12	11	pm5.74i 271	. . . . . . . 8 ⊢ ((𝑧 ∈ On → (Lim 𝑧 → 𝑥 ∈ 𝑧)) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
13	6, 7, 12	3bitri 297	. . . . . . 7 ⊢ ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
14		onsssuc 6409	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ suc 𝑥))
15		ontri1 6351	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
16	14, 15	bitr3d 281	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
17	16	ancoms 458	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥 ∈ 𝑧))
18		limeq 6329	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
19	18	notbid 318	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
20	19	elrab 3646	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧))
21	20	a1i 11	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
22	17, 21	imbi12d 344	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → ((𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
23	22	pm5.74da 803	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥 ∈ 𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
24	13, 23	bitr4id 290	. . . . . 6 ⊢ (𝑥 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))))
25		impexp 450	. . . . . . 7 ⊢ (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26		simpr 484	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27		onsuc 7755	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
28		onelon 6342	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
29	28	ex 412	. . . . . . . . . . 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 551	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → (𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥)))
32	26, 31	impbid2 226	. . . . . . . 8 ⊢ (𝑥 ∈ On → ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) ↔ 𝑧 ∈ suc 𝑥))
33	32	imbi1d 341	. . . . . . 7 ⊢ (𝑥 ∈ On → (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
34	25, 33	bitr3id 285	. . . . . 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 279	. . . . 5 ⊢ (𝑥 ∈ On → ((Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ (𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
36	35	albidv 1921	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37		df-ss 3918	. . . 4 ⊢ (suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ ∀𝑧(𝑧 ∈ suc 𝑥 → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
38	36, 37	bitr4di 289	. . 3 ⊢ (𝑥 ∈ On → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) ↔ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
39	38	rabbiia 3403	. 2 ⊢ {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
40	1, 39	eqtri 2759	1 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ⊆ wss 3901  Oncon0 6317  Lim wlim 6318  suc csuc 6319  ωcom 7808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-om 7809

This theorem is referenced by: (None)