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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 4682 | . . 3 ⊢ (Tr ω ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)) | |
2 | onelon 5665 | . . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
3 | 2 | expcom 450 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ On)) |
4 | limord 5701 | . . . . . . . . . . . 12 ⊢ (Lim 𝑧 → Ord 𝑧) | |
5 | ordtr 5654 | . . . . . . . . . . . 12 ⊢ (Ord 𝑧 → Tr 𝑧) | |
6 | trel 4687 | . . . . . . . . . . . 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 451 | . . . . . . . . . 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 1832 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) → ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) |
12 | 3, 11 | anim12d 584 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))) |
13 | elom 6960 | . . . . . 6 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧))) | |
14 | elom 6960 | . . . . . 6 ⊢ (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) | |
15 | 12, 13, 14 | 3imtr4g 284 | . . . . 5 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω)) |
16 | 15 | imp 444 | . . . 4 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
17 | 16 | ax-gen 1713 | . . 3 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
18 | 1, 17 | mpgbir 1717 | . 2 ⊢ Tr ω |
19 | omsson 6961 | . 2 ⊢ ω ⊆ On | |
20 | ordon 6874 | . 2 ⊢ Ord On | |
21 | trssord 5657 | . 2 ⊢ ((Tr ω ∧ ω ⊆ On ∧ Ord On) → Ord ω) | |
22 | 18, 19, 20, 21 | mp3an 1416 | 1 ⊢ Ord ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 ⊆ wss 3540 Tr wtr 4680 Ord word 5639 Oncon0 5640 Lim wlim 5641 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: elnn 6967 omon 6968 limom 6972 ssnlim 6975 omsinds 6976 peano5 6981 nnarcl 7583 nnawordex 7604 oaabslem 7610 oaabs2 7612 omabslem 7613 onomeneq 8035 ominf 8057 findcard3 8088 nnsdomg 8104 dffi3 8220 wofib 8333 alephgeom 8788 iscard3 8799 iunfictbso 8820 unctb 8910 ackbij2lem1 8924 ackbij1lem3 8927 ackbij1lem18 8942 ackbij2 8948 cflim2 8968 fin23lem26 9030 fin23lem23 9031 fin23lem27 9033 fin67 9100 alephexp1 9280 pwfseqlem3 9361 pwcdandom 9368 winainflem 9394 wunex2 9439 om2uzoi 12616 ltweuz 12622 fz1isolem 13102 mreexexdOLD 16132 1stcrestlem 21065 hfuni 31461 hfninf 31463 finxpreclem4 32407 |
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