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Theorem limom 7903
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
StepHypRef Expression
1 ordom 7897 . 2 Ord ω
2 ordeleqon 7802 . . 3 (Ord ω ↔ (ω ∈ On ∨ ω = On))
3 ordirr 6402 . . . . . . 7 (Ord ω → ¬ ω ∈ ω)
41, 3ax-mp 5 . . . . . 6 ¬ ω ∈ ω
5 elom 7890 . . . . . . 7 (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
65baib 535 . . . . . 6 (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
74, 6mtbii 326 . . . . 5 (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8 limomss 7892 . . . . . . . . . . 11 (Lim 𝑥 → ω ⊆ 𝑥)
9 limord 6444 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
10 ordsseleq 6413 . . . . . . . . . . . 12 ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
111, 9, 10sylancr 587 . . . . . . . . . . 11 (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
128, 11mpbid 232 . . . . . . . . . 10 (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
1312ord 865 . . . . . . . . 9 (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14 limeq 6396 . . . . . . . . . 10 (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
1514biimprcd 250 . . . . . . . . 9 (Lim 𝑥 → (ω = 𝑥 → Lim ω))
1613, 15syld 47 . . . . . . . 8 (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
1716con1d 145 . . . . . . 7 (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
1817com12 32 . . . . . 6 (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
1918alrimiv 1927 . . . . 5 (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
207, 19nsyl2 141 . . . 4 (ω ∈ On → Lim ω)
21 limon 7856 . . . . 5 Lim On
22 limeq 6396 . . . . 5 (ω = On → (Lim ω ↔ Lim On))
2321, 22mpbiri 258 . . . 4 (ω = On → Lim ω)
2420, 23jaoi 858 . . 3 ((ω ∈ On ∨ ω = On) → Lim ω)
252, 24sylbi 217 . 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 206  wo 848  wal 1538   = wceq 1540  wcel 2108  wss 3951  Ord word 6383  Oncon0 6384  Lim wlim 6385  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888
This theorem is referenced by:  peano2b  7904  ssnlim  7907  peano1OLD  7911  onesuc  8568  oaabslem  8685  oaabs2  8687  omabslem  8688  infensuc  9195  infeq5i  9676  elom3  9688  omenps  9695  omensuc  9696  infdifsn  9697  cardlim  10012  r1om  10283  cfom  10304  ominf4  10352  alephom  10625  wunex3  10781  satom  35361  fmla  35386  exrecfnlem  37380  onexlimgt  43255  oaabsb  43307  nnoeomeqom  43325  succlg  43341  dflim5  43342  dfom6  43544
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