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Mirrors > Home > ILE Home > Th. List > ordelord | GIF version |
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
Ref | Expression |
---|---|
ordelord | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2142 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | anbi2d 452 | . . . 4 ⊢ (𝑥 = 𝐵 → ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (Ord 𝐴 ∧ 𝐵 ∈ 𝐴))) |
3 | ordeq 4135 | . . . 4 ⊢ (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵)) | |
4 | 2, 3 | imbi12d 232 | . . 3 ⊢ (𝑥 = 𝐵 → (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) ↔ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵))) |
5 | dford3 4130 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
6 | 5 | simprbi 269 | . . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
7 | 6 | r19.21bi 2450 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Tr 𝑥) |
8 | ordelss 4142 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴) | |
9 | simpl 107 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴) | |
10 | trssord 4143 | . . . 4 ⊢ ((Tr 𝑥 ∧ 𝑥 ⊆ 𝐴 ∧ Ord 𝐴) → Ord 𝑥) | |
11 | 7, 8, 9, 10 | syl3anc 1170 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) |
12 | 4, 11 | vtoclg 2659 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)) |
13 | 12 | anabsi7 546 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ∀wral 2349 ⊆ wss 2974 Tr wtr 3883 Ord word 4125 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-in 2980 df-ss 2987 df-uni 3610 df-tr 3884 df-iord 4129 |
This theorem is referenced by: tron 4145 ordelon 4146 ordsucg 4254 ordwe 4326 smores 5941 |
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