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Theorem limomss 7804
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 6376 . 2 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7713 . . 3 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 elom 7802 . . . . . . . . . 10 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
43simprbi 497 . . . . . . . . 9 (𝑥 ∈ ω → ∀𝑦(Lim 𝑦𝑥𝑦))
5 limeq 6328 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6 eleq2 2826 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
75, 6imbi12d 344 . . . . . . . . . 10 (𝑦 = 𝐴 → ((Lim 𝑦𝑥𝑦) ↔ (Lim 𝐴𝑥𝐴)))
87spcgv 3554 . . . . . . . . 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 407 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥𝐴))
1211ssrdv 3949 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
1312ex 413 . . . 4 (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14 omsson 7803 . . . . . 6 ω ⊆ On
15 sseq2 3969 . . . . . 6 (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
1614, 15mpbiri 257 . . . . 5 (𝐴 = On → ω ⊆ 𝐴)
1716a1d 25 . . . 4 (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
1813, 17jaoi 855 . . 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 396  wo 845  wal 1539   = wceq 1541  wcel 2106  wss 3909  Ord word 6315  Oncon0 6316  Lim wlim 6317  ωcom 7799
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-tr 5222  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-ord 6319  df-on 6320  df-lim 6321  df-om 7800
This theorem is referenced by:  limom  7815  rdg0  8364  frfnom  8378  frsuc  8380  r1fin  9706  rankdmr1  9734  rankeq0b  9793  cardlim  9905  ackbij2  10176  cfom  10197  wunom  10653  inar1  10708  bj-rdg0gALT  35531
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