Theorem limomss 7916

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 6450	. 2 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7830	. . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		elom 7914	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
4	3	simprbi 503	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
5		limeq 6400	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6		eleq2 2853	. . . . . . . . . . 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 412	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴))
12	11	ssrdv 3919	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
13	12	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14		omsson 7915	. . . . . 6 ⊢ ω ⊆ On
15		sseq2 3939	. . . . . 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 2147   ⊆ wss 3900  Ord word 6387  Oncon0 6388  Lim wlim 6389  ωcom 7911

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393  df-om 7912

This theorem is used by:  limom  7927  rdg0  8467  frfnom  8481  frsuc  8483  r1fin  9817  rankdmr1  9846  rankeq0b  9913  cardlim  10090  ackbij2  10359  cfom  10381  wunom  10844  inar1  10899  bj-rdg0gALT  38256