Theorem limomss 7865

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 6422	. 2 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7779	. . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		elom 7863	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
4	3	simprbi 502	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
5		limeq 6372	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6		eleq2 2851	. . . . . . . . . . 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 3554	. . . . . . . . 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 411	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴))
12	11	ssrdv 3942	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
13	12	ex 417	. . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14		omsson 7864	. . . . . 6 ⊢ ω ⊆ On
15		sseq2 3962	. . . . . 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 870	. . 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 400   ∨ wo 860  ∀wal 1567   = wceq 1569   ∈ wcel 2142   ⊆ wss 3904  Ord word 6359  Oncon0 6360  Lim wlim 6361  ωcom 7860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-lim 6365  df-om 7861

This theorem is used by:  limom  7876  rdg0  8406  frfnom  8420  frsuc  8422  r1fin  9743  rankdmr1  9771  rankeq0b  9830  cardlim  9965  ackbij2  10232  cfom  10254  wunom  10711  inar1  10766  bj-rdg0gALT  37735