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Theorem limom 6949
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 6943 . 2 Ord ω
2 ordeleqon 6857 . . 3 (Ord ω ↔ (ω ∈ On ∨ ω = On))
3 ordirr 5644 . . . . . . 7 (Ord ω → ¬ ω ∈ ω)
41, 3ax-mp 5 . . . . . 6 ¬ ω ∈ ω
5 elom 6937 . . . . . . 7 (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
65baib 941 . . . . . 6 (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)))
74, 6mtbii 314 . . . . 5 (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
8 limomss 6939 . . . . . . . . . . 11 (Lim 𝑥 → ω ⊆ 𝑥)
9 limord 5687 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
10 ordsseleq 5655 . . . . . . . . . . . 12 ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
111, 9, 10sylancr 693 . . . . . . . . . . 11 (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥)))
128, 11mpbid 220 . . . . . . . . . 10 (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥))
1312ord 390 . . . . . . . . 9 (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥))
14 limeq 5638 . . . . . . . . . 10 (ω = 𝑥 → (Lim ω ↔ Lim 𝑥))
1514biimprcd 238 . . . . . . . . 9 (Lim 𝑥 → (ω = 𝑥 → Lim ω))
1613, 15syld 45 . . . . . . . 8 (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω))
1716con1d 137 . . . . . . 7 (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥))
1817com12 32 . . . . . 6 (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥))
1918alrimiv 1841 . . . . 5 (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))
207, 19nsyl2 140 . . . 4 (ω ∈ On → Lim ω)
21 limon 6905 . . . . 5 Lim On
22 limeq 5638 . . . . 5 (ω = On → (Lim ω ↔ Lim On))
2321, 22mpbiri 246 . . . 4 (ω = On → Lim ω)
2420, 23jaoi 392 . . 3 ((ω ∈ On ∨ ω = On) → Lim ω)
252, 24sylbi 205 . 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 194  wo 381  wal 1472   = wceq 1474  wcel 1976  wss 3539  Ord word 5625  Oncon0 5626  Lim wlim 5627  ωcom 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-om 6935
This theorem is referenced by:  peano2b  6950  ssnlim  6952  peano1  6954  onesuc  7474  oaabslem  7587  oaabs2  7589  omabslem  7590  infensuc  8000  infeq5i  8393  elom3  8405  omenps  8412  omensuc  8413  infdifsn  8414  cardlim  8658  r1om  8926  cfom  8946  ominf4  8994  alephom  9263  wunex3  9419
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