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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 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
Distinct variable group:   𝑥,𝐴

Proof of Theorem ssnlim
StepHypRef Expression
1 limom 7867 . . . 4 Lim ω
2 ssel 3970 . . . . 5 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥}))
3 limeq 6369 . . . . . . . 8 (𝑥 = ω → (Lim 𝑥 ↔ Lim ω))
43notbid 318 . . . . . . 7 (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω))
54elrab 3678 . . . . . 6 (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω))
65simprbi 496 . . . . 5 (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ Lim ω)
72, 6syl6 35 . . . 4 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ¬ Lim ω))
81, 7mt2i 137 . . 3 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ ω ∈ 𝐴)
98adantl 481 . 2 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴)
10 ordom 7861 . . . 4 Ord ω
11 ordtri1 6390 . . . 4 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
1210, 11mpan2 688 . . 3 (Ord 𝐴 → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
1312adantr 480 . 2 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
149, 13mpbird 257 1 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {crab 3426  wss 3943  Ord word 6356  Oncon0 6357  Lim wlim 6358  ωcom 7851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-om 7852
This theorem is referenced by: (None)
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