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Theorem ssnlim 7584
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 7580 . . . 4 Lim ω
2 ssel 3935 . . . . 5 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥}))
3 limeq 6181 . . . . . . . 8 (𝑥 = ω → (Lim 𝑥 ↔ Lim ω))
43notbid 321 . . . . . . 7 (𝑥 = ω → (¬ Lim 𝑥 ↔ ¬ Lim ω))
54elrab 3655 . . . . . 6 (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ∈ On ∧ ¬ Lim ω))
65simprbi 500 . . . . 5 (ω ∈ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ Lim ω)
72, 6syl6 35 . . . 4 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → (ω ∈ 𝐴 → ¬ Lim ω))
81, 7mt2i 139 . . 3 (𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} → ¬ ω ∈ 𝐴)
98adantl 485 . 2 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → ¬ ω ∈ 𝐴)
10 ordom 7574 . . . 4 Ord ω
11 ordtri1 6202 . . . 4 ((Ord 𝐴 ∧ Ord ω) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
1210, 11mpan2 690 . . 3 (Ord 𝐴 → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
1312adantr 484 . 2 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → (𝐴 ⊆ ω ↔ ¬ ω ∈ 𝐴))
149, 13mpbird 260 1 ((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  {crab 3134  wss 3908  Ord word 6168  Oncon0 6169  Lim wlim 6170  ωcom 7565
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-om 7566
This theorem is referenced by: (None)
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