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Theorem limomss 6962
 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 5701 . 2 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 6880 . . 3 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 elom 6960 . . . . . . . . . 10 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
43simprbi 479 . . . . . . . . 9 (𝑥 ∈ ω → ∀𝑦(Lim 𝑦𝑥𝑦))
5 limeq 5652 . . . . . . . . . . 11 (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6 eleq2 2677 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
75, 6imbi12d 333 . . . . . . . . . 10 (𝑦 = 𝐴 → ((Lim 𝑦𝑥𝑦) ↔ (Lim 𝐴𝑥𝐴)))
87spcgv 3266 . . . . . . . . 9 (𝐴 ∈ On → (∀𝑦(Lim 𝑦𝑥𝑦) → (Lim 𝐴𝑥𝐴)))
94, 8syl5 33 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ ω → (Lim 𝐴𝑥𝐴)))
109com23 84 . . . . . . 7 (𝐴 ∈ On → (Lim 𝐴 → (𝑥 ∈ ω → 𝑥𝐴)))
1110imp 444 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥𝐴))
1211ssrdv 3574 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
1312ex 449 . . . 4 (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14 omsson 6961 . . . . . 6 ω ⊆ On
15 sseq2 3590 . . . . . 6 (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
1614, 15mpbiri 247 . . . . 5 (𝐴 = On → ω ⊆ 𝐴)
1716a1d 25 . . . 4 (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
1813, 17jaoi 393 . . 3 ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
192, 18sylbi 206 . 2 (Ord 𝐴 → (Lim 𝐴 → ω ⊆ 𝐴))
201, 19mpcom 37 1 (Lim 𝐴 → ω ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639  Oncon0 5640  Lim wlim 5641  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958 This theorem is referenced by:  limom  6972  rdg0  7404  frfnom  7417  frsuc  7419  r1fin  8519  rankdmr1  8547  rankeq0b  8606  cardlim  8681  ackbij2  8948  cfom  8969  wunom  9421  inar1  9476
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