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Theorem limom 6972
 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
StepHypRef Expression
1 ordom 6966 . 2 Ord ω
2 ordeleqon 6880 . . 3 (Ord ω ↔ (ω ∈ On ∨ ω = On))
3 ordirr 5658 . . . . . . 7 (Ord ω → ¬ ω ∈ ω)
41, 3ax-mp 5 . . . . . 6 ¬ ω ∈ ω
5 elom 6960 . . . . . . 7 (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
65baib 942 . . . . . 6 (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
74, 6mtbii 315 . . . . 5 (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8 limomss 6962 . . . . . . . . . . 11 (Lim 𝑥 → ω ⊆ 𝑥)
9 limord 5701 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
10 ordsseleq 5669 . . . . . . . . . . . 12 ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
111, 9, 10sylancr 694 . . . . . . . . . . 11 (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
128, 11mpbid 221 . . . . . . . . . 10 (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
1312ord 391 . . . . . . . . 9 (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14 limeq 5652 . . . . . . . . . 10 (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
1514biimprcd 239 . . . . . . . . 9 (Lim 𝑥 → (ω = 𝑥 → Lim ω))
1613, 15syld 46 . . . . . . . 8 (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
1716con1d 138 . . . . . . 7 (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
1817com12 32 . . . . . 6 (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
1918alrimiv 1842 . . . . 5 (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
207, 19nsyl2 141 . . . 4 (ω ∈ On → Lim ω)
21 limon 6928 . . . . 5 Lim On
22 limeq 5652 . . . . 5 (ω = On → (Lim ω ↔ Lim On))
2321, 22mpbiri 247 . . . 4 (ω = On → Lim ω)
2420, 23jaoi 393 . . 3 ((ω ∈ On ∨ ω = On) → Lim ω)
252, 24sylbi 206 . 2 (Ord ω → Lim ω)
261, 25ax-mp 5 1 Lim ω
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639  Oncon0 5640  Lim wlim 5641  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958 This theorem is referenced by:  peano2b  6973  ssnlim  6975  peano1  6977  onesuc  7497  oaabslem  7610  oaabs2  7612  omabslem  7613  infensuc  8023  infeq5i  8416  elom3  8428  omenps  8435  omensuc  8436  infdifsn  8437  cardlim  8681  r1om  8949  cfom  8969  ominf4  9017  alephom  9286  wunex3  9442
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