Theorem limom 7597

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 7591	. 2 ⊢ Ord ω
2		ordeleqon 7505	. . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On))
3		ordirr 6211	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
4	1, 3	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
5		elom 7585	. . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
6	5	baib 538	. . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
7	4, 6	mtbii 328	. . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8		limomss 7587	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥)
9		limord 6252	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
10		ordsseleq 6222	. . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
11	1, 9, 10	sylancr 589	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
12	8, 11	mpbid 234	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
13	12	ord 860	. . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14		limeq 6205	. . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
15	14	biimprcd 252	. . . . . . . . 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 1928	. . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
20	7, 19	nsyl2 143	. . . 4 ⊢ (ω ∈ On → Lim ω)
21		limon 7553	. . . . 5 ⊢ Lim On
22		limeq 6205	. . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On))
23	21, 22	mpbiri 260	. . . 4 ⊢ (ω = On → Lim ω)
24	20, 23	jaoi 853	. . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω)
25	2, 24	sylbi 219	. 2 ⊢ (Ord ω → Lim ω)
26	1, 25	ax-mp 5	1 ⊢ Lim ω

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843  ∀wal 1535   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  Ord word 6192  Oncon0 6193  Lim wlim 6194  ωcom 7582

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-om 7583

This theorem is referenced by:  peano2b  7598  ssnlim  7601  peano1  7603  onesuc  8157  oaabslem  8272  oaabs2  8274  omabslem  8275  infensuc  8697  infeq5i  9101  elom3  9113  omenps  9120  omensuc  9121  infdifsn  9122  cardlim  9403  r1om  9668  cfom  9688  ominf4  9736  alephom  10009  wunex3  10165  satom  32605  fmla  32630  exrecfnlem  34662  dfom6  39905