Theorem ssnlim 2407

Description: An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42.

Assertion

Ref	Expression
ssnlim	⊢ ((Ord A ∧ A ⊆ {x ∈ On∣ ¬ Lim x}) → A ⊆ ω)

Distinct variable group(s):   x,A

Proof of Theorem ssnlim

Step	Hyp	Ref	Expression
1		limom 2387	. . . 4 ⊢ Lim ω
2		ssel 1502	. . . . 5 ⊢ (A ⊆ {x ∈ On∣ ¬ Lim x} → (ω ∈ A → ω ∈ {x ∈ On∣ ¬ Lim x}))
3		limeq 2211	. . . . . . . 8 ⊢ (x = ω → (Lim x ↔ Lim ω))
4	3	negbid 463	. . . . . . 7 ⊢ (x = ω → (¬ Lim x ↔ ¬ Lim ω))
5	4	elrab 1422	. . . . . 6 ⊢ (ω ∈ {x ∈ On∣ ¬ Lim x} ↔ (ω ∈ On ∧ ¬ Lim ω))
6	5	pm3.27bd 263	. . . . 5 ⊢ (ω ∈ {x ∈ On∣ ¬ Lim x} → ¬ Lim ω)
7	2, 6	syl6 23	. . . 4 ⊢ (A ⊆ {x ∈ On∣ ¬ Lim x} → (ω ∈ A → ¬ Lim ω))
8	1, 7	mt2i 97	. . 3 ⊢ (A ⊆ {x ∈ On∣ ¬ Lim x} → ¬ ω ∈ A)
9	8	adantl 305	. 2 ⊢ ((Ord A ∧ A ⊆ {x ∈ On∣ ¬ Lim x}) → ¬ ω ∈ A)
10		ordom 2382	. . . 4 ⊢ Ord ω
11		ordtri1 2231	. . . 4 ⊢ ((Ord A ∧ Ord ω) → (A ⊆ ω ↔ ¬ ω ∈ A))
12	10, 11	mpan2 519	. . 3 ⊢ (Ord A → (A ⊆ ω ↔ ¬ ω ∈ A))
13	12	adantr 306	. 2 ⊢ ((Ord A ∧ A ⊆ {x ∈ On∣ ¬ Lim x}) → (A ⊆ ω ↔ ¬ ω ∈ A))
14	9, 13	mpbird 171	1 ⊢ ((Ord A ∧ A ⊆ {x ∈ On∣ ¬ Lim x}) → A ⊆ ω)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204   ⊆ wss 1487  Ord word 2198  Oncon0 2199  Lim wlim 2200  ωcom 2372

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

metamath.org