Theorem limomss 7807

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.

Step	Hyp	Ref	Expression
1		limord 6373	. 2 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7721	. . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3		elom 7805	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
4	3	simprbi 496	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
5		limeq 6324	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴))
6		eleq2 2820	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴))
7	5, 6	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴)))
8	7	spcgv 3546	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) → (Lim 𝐴 → 𝑥 ∈ 𝐴)))
9	4, 8	syl5 34	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ ω → (Lim 𝐴 → 𝑥 ∈ 𝐴)))
10	9	com23 86	. . . . . . 7 ⊢ (𝐴 ∈ On → (Lim 𝐴 → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)))
11	10	imp 406	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴))
12	11	ssrdv 3935	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴)
13	12	ex 412	. . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴))
14		omsson 7806	. . . . . 6 ⊢ ω ⊆ On
15		sseq2 3956	. . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On))
16	14, 15	mpbiri 258	. . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴)
17	16	a1d 25	. . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴))
18	13, 17	jaoi 857	. . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴))
19	2, 18	sylbi 217	. 2 ⊢ (Ord 𝐴 → (Lim 𝐴 → ω ⊆ 𝐴))
20	1, 19	mpcom 38	1 ⊢ (Lim 𝐴 → ω ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  Ord word 6311  Oncon0 6312  Lim wlim 6313  ωcom 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6315  df-on 6316  df-lim 6317  df-om 7803

This theorem is referenced by:  limom  7818  rdg0  8346  frfnom  8360  frsuc  8362  r1fin  9672  rankdmr1  9700  rankeq0b  9759  cardlim  9871  ackbij2  10139  cfom  10161  wunom  10617  inar1  10672  bj-rdg0gALT  37122