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Theorem limomss 7812
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 6382 . 2 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7721 . . 3 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 elom 7810 . . . . . . . . . 10 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
43simprbi 498 . . . . . . . . 9 (𝑥 ∈ ω → ∀𝑦(Lim 𝑦𝑥𝑦))
5 limeq 6334 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6 eleq2 2827 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
75, 6imbi12d 345 . . . . . . . . . 10 (𝑦 = 𝐴 → ((Lim 𝑦𝑥𝑦) ↔ (Lim 𝐴𝑥𝐴)))
87spcgv 3558 . . . . . . . . 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 408 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥𝐴))
1211ssrdv 3955 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
1312ex 414 . . . 4 (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14 omsson 7811 . . . . . 6 ω ⊆ On
15 sseq2 3975 . . . . . 6 (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
1614, 15mpbiri 258 . . . . 5 (𝐴 = On → ω ⊆ 𝐴)
1716a1d 25 . . . 4 (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
1813, 17jaoi 856 . . 3 ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
192, 18sylbi 216 . 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 397  wo 846  wal 1540   = wceq 1542  wcel 2107  wss 3915  Ord word 6321  Oncon0 6322  Lim wlim 6323  ωcom 7807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-om 7808
This theorem is referenced by:  limom  7823  rdg0  8372  frfnom  8386  frsuc  8388  r1fin  9716  rankdmr1  9744  rankeq0b  9803  cardlim  9915  ackbij2  10186  cfom  10207  wunom  10663  inar1  10718  bj-rdg0gALT  35571
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