Theorem limom 7879

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 7873	. 2 ⊢ Ord ω
2		ordeleqon 7782	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordirr 6375	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
5		elom 7866	. . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
6	5	baib 545	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
7	4, 6	mtbii 329	. . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8		limomss 7868	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥)
9		limord 6419	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
10		ordsseleq 6387	. . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
11	1, 9, 10	sylancr 599	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
12	8, 11	mpbid 235	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
13	12	ord 878	. . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14		limeq 6369	. . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
15	14	biimprcd 253	. . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω))
16	13, 15	syld 48	. . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
17	16	con1d 146	. . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
18	17	com12 33	. . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
19	18	alrimiv 1960	. . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
20	7, 19	nsyl2 142	. . . 4 ⊢ (ω ∈ On → Lim ω)
21		limon 7833	. . . . 5 ⊢ Lim On
22		limeq 6369	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
23	21, 22	mpbiri 261	. . . 4 ⊢ (ω = On → Lim ω)
24	20, 23	jaoi 871	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
25	2, 24	sylbi 220	. 2 ⊢ (Ord ω → Lim ω)
26	1, 25	ax-mp 5	1 ⊢ Lim ω

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  Ord word 6356  Oncon0 6357  Lim wlim 6358  ωcom 7863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-om 7864

This theorem is used by:  peano2b  7880  ssnlim  7883  onesuc  8520  oaabslem  8638  oaabs2  8640  omabslem  8641  infensuc  9156  infeq5i  9618  elom3  9630  omenps  9637  omensuc  9638  infdifsn  9639  cardlim  9980  r1om  10248  cfom  10269  ominf4  10317  alephom  10597  wunex3  10753  r1omhf  35617  r1omfv  35621  satom  35938  fmla  35963  exrecfnlem  38136  onexlimgt  44087  oaabsb  44138  nnoeomeqom  44156  succlg  44172  dflim5  44173  dfom6  44374