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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 4527 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) | |
2 | 1 | gen2 1427 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) |
3 | dftr2 4036 | . . 3 ⊢ (Tr ω ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)) | |
4 | 2, 3 | mpbir 145 | . 2 ⊢ Tr ω |
5 | treq 4040 | . . . 4 ⊢ (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅)) | |
6 | treq 4040 | . . . 4 ⊢ (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥)) | |
7 | treq 4040 | . . . 4 ⊢ (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥)) | |
8 | tr0 4045 | . . . 4 ⊢ Tr ∅ | |
9 | suctr 4351 | . . . . 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 4522 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) |
12 | 11 | rgen 2488 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
13 | dford3 4297 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
14 | 4, 12, 13 | mpbir2an 927 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 ∈ wcel 1481 ∀wral 2417 ∅c0 3368 Tr wtr 4034 Ord word 4292 suc csuc 4295 ωcom 4512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-suc 4301 df-iom 4513 |
This theorem is referenced by: omelon2 4529 limom 4535 freccllem 6307 frecfcllem 6309 frecsuclem 6311 fict 6770 infnfi 6797 isinfinf 6799 hashinfuni 10555 hashinfom 10556 hashennn 10558 ennnfonelemrn 11968 ctinf 11979 |
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