Theorem limom 7703

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 7697	. 2 ⊢ Ord ω
2		ordeleqon 7609	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordirr 6269	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
5		elom 7690	. . . . . . 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 325	. . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8		limomss 7692	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥)
9		limord 6310	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
10		ordsseleq 6280	. . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
11	1, 9, 10	sylancr 586	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
12	8, 11	mpbid 231	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
13	12	ord 860	. . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14		limeq 6263	. . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
15	14	biimprcd 249	. . . . . . . . 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 1931	. . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
20	7, 19	nsyl2 141	. . . 4 ⊢ (ω ∈ On → Lim ω)
21		limon 7658	. . . . 5 ⊢ Lim On
22		limeq 6263	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
23	21, 22	mpbiri 257	. . . 4 ⊢ (ω = On → Lim ω)
24	20, 23	jaoi 853	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
25	2, 24	sylbi 216	. 2 ⊢ (Ord ω → Lim ω)
26	1, 25	ax-mp 5	1 ⊢ Lim ω

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843  ∀wal 1537   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  Ord word 6250  Oncon0 6251  Lim wlim 6252  ωcom 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-om 7688

This theorem is referenced by:  peano2b  7704  ssnlim  7707  peano1  7710  onesuc  8322  oaabslem  8437  oaabs2  8439  omabslem  8440  infensuc  8891  infeq5i  9324  elom3  9336  omenps  9343  omensuc  9344  infdifsn  9345  cardlim  9661  r1om  9931  cfom  9951  ominf4  9999  alephom  10272  wunex3  10428  satom  33218  fmla  33243  exrecfnlem  35477  dfom6  41036