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Mirrors > Home > ILE Home > Th. List > ordom | GIF version |
Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
ordom | ⊢ Ord ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn 4457 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) | |
2 | 1 | gen2 1394 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) |
3 | dftr2 3968 | . . 3 ⊢ (Tr ω ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)) | |
4 | 2, 3 | mpbir 145 | . 2 ⊢ Tr ω |
5 | treq 3972 | . . . 4 ⊢ (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅)) | |
6 | treq 3972 | . . . 4 ⊢ (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥)) | |
7 | treq 3972 | . . . 4 ⊢ (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥)) | |
8 | tr0 3977 | . . . 4 ⊢ Tr ∅ | |
9 | suctr 4281 | . . . . 5 ⊢ (Tr 𝑥 → Tr suc 𝑥) | |
10 | 9 | a1i 9 | . . . 4 ⊢ (𝑥 ∈ ω → (Tr 𝑥 → Tr suc 𝑥)) |
11 | 5, 6, 7, 6, 8, 10 | finds 4452 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) |
12 | 11 | rgen 2444 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
13 | dford3 4227 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
14 | 4, 12, 13 | mpbir2an 894 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1297 ∈ wcel 1448 ∀wral 2375 ∅c0 3310 Tr wtr 3966 Ord word 4222 suc csuc 4225 ωcom 4442 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-tr 3967 df-iord 4226 df-suc 4231 df-iom 4443 |
This theorem is referenced by: omelon2 4459 limom 4465 freccllem 6229 frecfcllem 6231 frecsuclem 6233 fict 6691 infnfi 6718 isinfinf 6720 hashinfuni 10364 hashinfom 10365 hashennn 10367 ennnfonelemrn 11724 ctinf 11735 |
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