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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 4519 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) | |
2 | 1 | gen2 1426 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) |
3 | dftr2 4028 | . . 3 ⊢ (Tr ω ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)) | |
4 | 2, 3 | mpbir 145 | . 2 ⊢ Tr ω |
5 | treq 4032 | . . . 4 ⊢ (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅)) | |
6 | treq 4032 | . . . 4 ⊢ (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥)) | |
7 | treq 4032 | . . . 4 ⊢ (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥)) | |
8 | tr0 4037 | . . . 4 ⊢ Tr ∅ | |
9 | suctr 4343 | . . . . 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 4514 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) |
12 | 11 | rgen 2485 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
13 | dford3 4289 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
14 | 4, 12, 13 | mpbir2an 926 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 ∈ wcel 1480 ∀wral 2416 ∅c0 3363 Tr wtr 4026 Ord word 4284 suc csuc 4287 ωcom 4504 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-tr 4027 df-iord 4288 df-suc 4293 df-iom 4505 |
This theorem is referenced by: omelon2 4521 limom 4527 freccllem 6299 frecfcllem 6301 frecsuclem 6303 fict 6762 infnfi 6789 isinfinf 6791 hashinfuni 10523 hashinfom 10524 hashennn 10526 ennnfonelemrn 11932 ctinf 11943 |
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