Theorem limomss 7883

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 6424	. 2 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7797	. . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		elom 7881	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
4	3	simprbi 503	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
5		limeq 6374	. . . . . . . . . . 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 3914	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
13	12	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14		omsson 7882	. . . . . 6 ⊢ ω ⊆ On
15		sseq2 3934	. . . . . 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 1567   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  Ord word 6361  Oncon0 6362  Lim wlim 6363  ωcom 7878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-lim 6367  df-om 7879

This theorem is used by:  limom  7894  rdg0  8433  frfnom  8447  frsuc  8449  r1fin  9784  rankdmr1  9813  rankeq0b  9880  cardlim  10057  ackbij2  10324  cfom  10346  wunom  10809  inar1  10864  bj-rdg0gALT  38079