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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 4617 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) | |
2 | 1 | gen2 1460 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω) |
3 | dftr2 4115 | . . 3 ⊢ (Tr ω ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)) | |
4 | 2, 3 | mpbir 146 | . 2 ⊢ Tr ω |
5 | treq 4119 | . . . 4 ⊢ (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅)) | |
6 | treq 4119 | . . . 4 ⊢ (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥)) | |
7 | treq 4119 | . . . 4 ⊢ (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥)) | |
8 | tr0 4124 | . . . 4 ⊢ Tr ∅ | |
9 | suctr 4433 | . . . . 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 4611 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) |
12 | 11 | rgen 2540 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
13 | dford3 4379 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
14 | 4, 12, 13 | mpbir2an 943 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 ∈ wcel 2158 ∀wral 2465 ∅c0 3434 Tr wtr 4113 Ord word 4374 suc csuc 4377 ωcom 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-tr 4114 df-iord 4378 df-suc 4383 df-iom 4602 |
This theorem is referenced by: omelon2 4619 limom 4625 freccllem 6417 frecfcllem 6419 frecsuclem 6421 fict 6882 infnfi 6909 isinfinf 6911 hashinfuni 10771 hashinfom 10772 hashennn 10774 ennnfonelemrn 12434 ctinf 12445 |
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