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Theorem limom 7247
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 7241 . 2 Ord ω
2 ordeleqon 7155 . . 3 (Ord ω ↔ (ω ∈ On ∨ ω = On))
3 ordirr 5903 . . . . . . 7 (Ord ω → ¬ ω ∈ ω)
41, 3ax-mp 5 . . . . . 6 ¬ ω ∈ ω
5 elom 7235 . . . . . . 7 (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
65baib 982 . . . . . 6 (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
74, 6mtbii 315 . . . . 5 (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8 limomss 7237 . . . . . . . . . . 11 (Lim 𝑥 → ω ⊆ 𝑥)
9 limord 5946 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
10 ordsseleq 5914 . . . . . . . . . . . 12 ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
111, 9, 10sylancr 698 . . . . . . . . . . 11 (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
128, 11mpbid 222 . . . . . . . . . 10 (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
1312ord 391 . . . . . . . . 9 (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14 limeq 5897 . . . . . . . . . 10 (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
1514biimprcd 240 . . . . . . . . 9 (Lim 𝑥 → (ω = 𝑥 → Lim ω))
1613, 15syld 47 . . . . . . . 8 (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
1716con1d 139 . . . . . . 7 (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
1817com12 32 . . . . . 6 (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
1918alrimiv 2005 . . . . 5 (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
207, 19nsyl2 142 . . . 4 (ω ∈ On → Lim ω)
21 limon 7203 . . . . 5 Lim On
22 limeq 5897 . . . . 5 (ω = On → (Lim ω ↔ Lim On))
2321, 22mpbiri 248 . . . 4 (ω = On → Lim ω)
2420, 23jaoi 393 . . 3 ((ω ∈ On ∨ ω = On) → Lim ω)
252, 24sylbi 207 . 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 196  wo 382  wal 1630   = wceq 1632  wcel 2140  wss 3716  Ord word 5884  Oncon0 5885  Lim wlim 5886  ωcom 7232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-om 7233
This theorem is referenced by:  peano2b  7248  ssnlim  7250  peano1  7252  onesuc  7782  oaabslem  7895  oaabs2  7897  omabslem  7898  infensuc  8306  infeq5i  8709  elom3  8721  omenps  8728  omensuc  8729  infdifsn  8730  cardlim  9009  r1om  9279  cfom  9299  ominf4  9347  alephom  9620  wunex3  9776
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