Theorem ssnlim 7811

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 7807	. . . 4 ⊢ Lim ω
2		ssel 3923	. . . . 5 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥}))
3		limeq 6313	. . . . . . . 8 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω))
4	3	notbid 318	. . . . . . 7 ⊢ (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω))
5	4	elrab 3642	. . . . . 6 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω))
6	5	simprbi 496	. . . . 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 481	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴)
10		ordom 7801	. . . 4 ⊢ Ord ω
11		ordtri1 6334	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
12	10, 11	mpan2 691	. . 3 ⊢ (Ord 𝐴 → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
13	12	adantr 480	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
14	9, 13	mpbird 257	1 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3897  Ord word 6300  Oncon0 6301  Lim wlim 6302  ωcom 7791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-om 7792

This theorem is referenced by: (None)