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Mirrors > Home > MPE Home > Th. List > ordom | Structured version Visualization version GIF version |
Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ordom | ⊢ Ord ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 5165 | . . 3 ⊢ (Tr ω ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)) | |
2 | onelon 6209 | . . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
3 | 2 | expcom 414 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ On)) |
4 | limord 6243 | . . . . . . . . . . . 12 ⊢ (Lim 𝑧 → Ord 𝑧) | |
5 | ordtr 6198 | . . . . . . . . . . . 12 ⊢ (Ord 𝑧 → Tr 𝑧) | |
6 | trel 5170 | . . . . . . . . . . . 12 ⊢ (Tr 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧)) | |
7 | 4, 5, 6 | 3syl 18 | . . . . . . . . . . 11 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧)) |
8 | 7 | expd 416 | . . . . . . . . . 10 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))) |
9 | 8 | com12 32 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → (Lim 𝑧 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))) |
10 | 9 | a2d 29 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ((Lim 𝑧 → 𝑥 ∈ 𝑧) → (Lim 𝑧 → 𝑦 ∈ 𝑧))) |
11 | 10 | alimdv 1908 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) → ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) |
12 | 3, 11 | anim12d 608 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))) |
13 | elom 7572 | . . . . . 6 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧))) | |
14 | elom 7572 | . . . . . 6 ⊢ (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) | |
15 | 12, 13, 14 | 3imtr4g 297 | . . . . 5 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω)) |
16 | 15 | imp 407 | . . . 4 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
17 | 16 | ax-gen 1787 | . . 3 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
18 | 1, 17 | mpgbir 1791 | . 2 ⊢ Tr ω |
19 | omsson 7573 | . 2 ⊢ ω ⊆ On | |
20 | ordon 7487 | . 2 ⊢ Ord On | |
21 | trssord 6201 | . 2 ⊢ ((Tr ω ∧ ω ⊆ On ∧ Ord On) → Ord ω) | |
22 | 18, 19, 20, 21 | mp3an 1452 | 1 ⊢ Ord ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 ⊆ wss 3933 Tr wtr 5163 Ord word 6183 Oncon0 6184 Lim wlim 6185 ωcom 7569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 |
This theorem is referenced by: elnn 7579 omon 7580 limom 7584 ssnlim 7588 omsinds 7589 peano5 7594 omsucelsucb 8083 nnarcl 8231 nnawordex 8252 oaabslem 8259 oaabs2 8261 omabslem 8262 onomeneq 8696 ominf 8718 findcard3 8749 nnsdomg 8765 dffi3 8883 wofib 8997 alephgeom 9496 iscard3 9507 iunfictbso 9528 unctb 9615 ackbij2lem1 9629 ackbij1lem3 9632 ackbij1lem18 9647 ackbij2 9653 cflim2 9673 fin23lem26 9735 fin23lem23 9736 fin23lem27 9738 fin67 9805 alephexp1 9989 pwfseqlem3 10070 pwdjundom 10077 winainflem 10103 wunex2 10148 om2uzoi 13311 ltweuz 13317 fz1isolem 13807 1stcrestlem 21988 satfn 32499 hfuni 33542 hfninf 33544 finxpreclem4 34557 |
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