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Theorem limomss 7892
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 6444 . 2 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7802 . . 3 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 elom 7890 . . . . . . . . . 10 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
43simprbi 496 . . . . . . . . 9 (𝑥 ∈ ω → ∀𝑦(Lim 𝑦𝑥𝑦))
5 limeq 6396 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6 eleq2 2830 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
75, 6imbi12d 344 . . . . . . . . . 10 (𝑦 = 𝐴 → ((Lim 𝑦𝑥𝑦) ↔ (Lim 𝐴𝑥𝐴)))
87spcgv 3596 . . . . . . . . 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 406 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥𝐴))
1211ssrdv 3989 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
1312ex 412 . . . 4 (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14 omsson 7891 . . . . . 6 ω ⊆ On
15 sseq2 4010 . . . . . 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 858 . . 3 ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
192, 18sylbi 217 . 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 395  wo 848  wal 1538   = wceq 1540  wcel 2108  wss 3951  Ord word 6383  Oncon0 6384  Lim wlim 6385  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-om 7888
This theorem is referenced by:  limom  7903  rdg0  8461  frfnom  8475  frsuc  8477  r1fin  9813  rankdmr1  9841  rankeq0b  9900  cardlim  10012  ackbij2  10282  cfom  10304  wunom  10760  inar1  10815  bj-rdg0gALT  37072
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