Theorem ssnlim 7083

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 7080	. . . 4 ⊢ Lim ω
2		ssel 3597	. . . . 5 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥}))
3		limeq 5735	. . . . . . . 8 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω))
4	3	notbid 308	. . . . . . 7 ⊢ (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω))
5	4	elrab 3363	. . . . . 6 ⊢ (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω))
6	5	simprbi 480	. . . . 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 132	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ ω ∈ 𝐴)
9	8	adantl 482	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴)
10		ordom 7074	. . . 4 ⊢ Ord ω
11		ordtri1 5756	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
12	10, 11	mpan2 707	. . 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 247	1 ⊢ ((Ord 𝐴 ∧ 𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  Ord word 5722  Oncon0 5723  Lim wlim 5724  ωcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by: (None)