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Mirrors > Home > MPE Home > Th. List > limom | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
limom | ⊢ Lim ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7865 | . 2 ⊢ Ord ω | |
2 | ordeleqon 7769 | . . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
3 | ordirr 6383 | . . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω) | |
4 | 1, 3 | ax-mp 5 | . . . . . 6 ⊢ ¬ ω ∈ ω |
5 | elom 7858 | . . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) | |
6 | 5 | baib 537 | . . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) |
7 | 4, 6 | mtbii 326 | . . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
8 | limomss 7860 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥) | |
9 | limord 6425 | . . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥) | |
10 | ordsseleq 6394 | . . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) | |
11 | 1, 9, 10 | sylancr 588 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) |
12 | 8, 11 | mpbid 231 | . . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥)) |
13 | 12 | ord 863 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥)) |
14 | limeq 6377 | . . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥)) | |
15 | 14 | biimprcd 249 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω)) |
16 | 13, 15 | syld 47 | . . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω)) |
17 | 16 | con1d 145 | . . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥)) |
18 | 17 | com12 32 | . . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥)) |
19 | 18 | alrimiv 1931 | . . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
20 | 7, 19 | nsyl2 141 | . . . 4 ⊢ (ω ∈ On → Lim ω) |
21 | limon 7824 | . . . . 5 ⊢ Lim On | |
22 | limeq 6377 | . . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On)) | |
23 | 21, 22 | mpbiri 258 | . . . 4 ⊢ (ω = On → Lim ω) |
24 | 20, 23 | jaoi 856 | . . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω) |
25 | 2, 24 | sylbi 216 | . 2 ⊢ (Ord ω → Lim ω) |
26 | 1, 25 | ax-mp 5 | 1 ⊢ Lim ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 Ord word 6364 Oncon0 6365 Lim wlim 6366 ωcom 7855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-om 7856 |
This theorem is referenced by: peano2b 7872 ssnlim 7875 peano1OLD 7880 onesuc 8530 oaabslem 8646 oaabs2 8648 omabslem 8649 infensuc 9155 infeq5i 9631 elom3 9643 omenps 9650 omensuc 9651 infdifsn 9652 cardlim 9967 r1om 10239 cfom 10259 ominf4 10307 alephom 10580 wunex3 10736 satom 34347 fmla 34372 exrecfnlem 36260 onexlimgt 41992 oaabsb 42044 nnoeomeqom 42062 succlg 42078 dflim5 42079 dfom6 42282 |
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