Theorem limom 7442

Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. 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 7436	. 2 ⊢ Ord ω
2		ordeleqon 7350	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordirr 6076	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
5		elom 7430	. . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
6	5	baib 536	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
7	4, 6	mtbii 327	. . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8		limomss 7432	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥)
9		limord 6117	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
10		ordsseleq 6087	. . . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
13	12	ord 859	. . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14		limeq 6070	. . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
15	14	biimprcd 251	. . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω))
16	13, 15	syld 47	. . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
17	16	con1d 147	. . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
18	17	com12 32	. . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
19	18	alrimiv 1903	. . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
20	7, 19	nsyl2 143	. . . 4 ⊢ (ω ∈ On → Lim ω)
21		limon 7398	. . . . 5 ⊢ Lim On
22		limeq 6070	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
23	21, 22	mpbiri 259	. . . 4 ⊢ (ω = On → Lim ω)
24	20, 23	jaoi 852	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
25	2, 24	sylbi 218	. 2 ⊢ (Ord ω → Lim ω)
26	1, 25	ax-mp 5	1 ⊢ Lim ω

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∨ wo 842  ∀wal 1518   = wceq 1520   ∈ wcel 2079   ⊆ wss 3854  Ord word 6057  Oncon0 6058  Lim wlim 6059  ωcom 7427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-tr 5058  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-om 7428

This theorem is referenced by:  peano2b  7443  ssnlim  7446  peano1  7448  onesuc  7997  oaabslem  8111  oaabs2  8113  omabslem  8114  infensuc  8532  infeq5i  8934  elom3  8946  omenps  8953  omensuc  8954  infdifsn  8955  cardlim  9236  r1om  9501  cfom  9521  ominf4  9569  alephom  9842  wunex3  9998  satom  32167  fmla  32190  exrecfnlem  34137  dfom6  39334