Theorem ssnlim 7871

Description: An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem ssnlim

Step	Hyp	Ref	Expression
1		limom 7867	. . . 4 ⊢ Lim ω
2		ssel 3974	. . . . 5 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥}))
3		limeq 6373	. . . . . . . 8 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω))
4	3	notbid 317	. . . . . . 7 ⊢ (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω))
5	4	elrab 3682	. . . . . 6 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω))
6	5	simprbi 497	. . . . 5 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ Lim ω)
7	2, 6	syl6 35	. . . 4 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ¬ Lim ω))
8	1, 7	mt2i 137	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ ω ∈ 𝐴)
9	8	adantl 482	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴)
10		ordom 7861	. . . 4 ⊢ Ord ω
11		ordtri1 6394	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
12	10, 11	mpan2 689	. . 3 ⊢ (Ord 𝐴 → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
13	12	adantr 481	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
14	9, 13	mpbird 256	1 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   ⊆ wss 3947  Ord word 6360  Oncon0 6361  Lim wlim 6362  ωcom 7851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7852

This theorem is referenced by: (None)