Theorem limom 7903

Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Theorem 1.23 of [Schloeder] p. 4. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
limom	⊢ Lim ω

Proof of Theorem limom

Step	Hyp	Ref	Expression
1		ordom 7897	. 2 ⊢ Ord ω
2		ordeleqon 7801	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordirr 6404	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
5		elom 7890	. . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
6	5	baib 535	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
7	4, 6	mtbii 326	. . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8		limomss 7892	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥)
9		limord 6446	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
10		ordsseleq 6415	. . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
11	1, 9, 10	sylancr 587	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
12	8, 11	mpbid 232	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
13	12	ord 864	. . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14		limeq 6398	. . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
15	14	biimprcd 250	. . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω))
16	13, 15	syld 47	. . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
17	16	con1d 145	. . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
18	17	com12 32	. . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
19	18	alrimiv 1925	. . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
20	7, 19	nsyl2 141	. . . 4 ⊢ (ω ∈ On → Lim ω)
21		limon 7856	. . . . 5 ⊢ Lim On
22		limeq 6398	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
23	21, 22	mpbiri 258	. . . 4 ⊢ (ω = On → Lim ω)
24	20, 23	jaoi 857	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
25	2, 24	sylbi 217	. 2 ⊢ (Ord ω → Lim ω)
26	1, 25	ax-mp 5	1 ⊢ Lim ω

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847  ∀wal 1535   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  Ord word 6385  Oncon0 6386  Lim wlim 6387  ωcom 7887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888

This theorem is referenced by:  peano2b  7904  ssnlim  7907  peano1OLD  7912  onesuc  8567  oaabslem  8684  oaabs2  8686  omabslem  8687  infensuc  9194  infeq5i  9674  elom3  9686  omenps  9693  omensuc  9694  infdifsn  9695  cardlim  10010  r1om  10281  cfom  10302  ominf4  10350  alephom  10623  wunex3  10779  satom  35341  fmla  35366  exrecfnlem  37362  onexlimgt  43232  oaabsb  43284  nnoeomeqom  43302  succlg  43318  dflim5  43319  dfom6  43521