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Theorem limomss 7302
Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
limomss (Lim 𝐴 → ω ⊆ 𝐴)

Proof of Theorem limomss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limord 5998 . 2 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7220 . . 3 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 elom 7300 . . . . . . . . . 10 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
43simprbi 491 . . . . . . . . 9 (𝑥 ∈ ω → ∀𝑦(Lim 𝑦𝑥𝑦))
5 limeq 5951 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6 eleq2 2865 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
75, 6imbi12d 336 . . . . . . . . . 10 (𝑦 = 𝐴 → ((Lim 𝑦𝑥𝑦) ↔ (Lim 𝐴𝑥𝐴)))
87spcgv 3479 . . . . . . . . 9 (𝐴 ∈ On → (∀𝑦(Lim 𝑦𝑥𝑦) → (Lim 𝐴𝑥𝐴)))
94, 8syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ ω → (Lim 𝐴𝑥𝐴)))
109com23 86 . . . . . . 7 (𝐴 ∈ On → (Lim 𝐴 → (𝑥 ∈ ω → 𝑥𝐴)))
1110imp 396 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥𝐴))
1211ssrdv 3802 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
1312ex 402 . . . 4 (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14 omsson 7301 . . . . . 6 ω ⊆ On
15 sseq2 3821 . . . . . 6 (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
1614, 15mpbiri 250 . . . . 5 (𝐴 = On → ω ⊆ 𝐴)
1716a1d 25 . . . 4 (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
1813, 17jaoi 884 . . 3 ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
192, 18sylbi 209 . 2 (Ord 𝐴 → (Lim 𝐴 → ω ⊆ 𝐴))
201, 19mpcom 38 1 (Lim 𝐴 → ω ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874  wal 1651   = wceq 1653  wcel 2157  wss 3767  Ord word 5938  Oncon0 5939  Lim wlim 5940  ωcom 7297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-om 7298
This theorem is referenced by:  limom  7312  rdg0  7754  frfnom  7767  frsuc  7769  r1fin  8884  rankdmr1  8912  rankeq0b  8971  cardlim  9082  ackbij2  9351  cfom  9372  wunom  9828  inar1  9883
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