Theorem limomss 7867

Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
limomss	⊢ (Lim 𝐴 → ω ⊆ 𝐴)

Proof of Theorem limomss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limord 6419	. 2 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7781	. . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		elom 7865	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
4	3	simprbi 503	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
5		limeq 6369	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6		eleq2 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴))
7	5, 6	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴)))
8	7	spcgv 3550	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) → (Lim 𝐴 → 𝑥 ∈ 𝐴)))
9	4, 8	syl5 35	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ ω → (Lim 𝐴 → 𝑥 ∈ 𝐴)))
10	9	com23 87	. . . . . . 7 ⊢ (𝐴 ∈ On → (Lim 𝐴 → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)))
11	10	imp 412	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴))
12	11	ssrdv 3937	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
13	12	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14		omsson 7866	. . . . . 6 ⊢ ω ⊆ On
15		sseq2 3957	. . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
16	14, 15	mpbiri 261	. . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴)
17	16	a1d 26	. . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
18	13, 17	jaoi 871	. . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
19	2, 18	sylbi 220	. 2 ⊢ (Ord 𝐴 → (Lim 𝐴 → ω ⊆ 𝐴))
20	1, 19	mpcom 39	1 ⊢ (Lim 𝐴 → ω ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  Ord word 6356  Oncon0 6357  Lim wlim 6358  ωcom 7862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362  df-om 7863

This theorem is used by:  limom  7878  rdg0  8410  frfnom  8424  frsuc  8426  r1fin  9755  rankdmr1  9783  rankeq0b  9842  cardlim  9977  ackbij2  10244  cfom  10266  wunom  10729  inar1  10784  bj-rdg0gALT  37815