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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 2175 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | anbi2d 457 | . . . 4 ⊢ (𝑥 = 𝐵 → ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (Ord 𝐴 ∧ 𝐵 ∈ 𝐴))) |
3 | ordeq 4252 | . . . 4 ⊢ (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵)) | |
4 | 2, 3 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝐵 → (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) ↔ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵))) |
5 | dford3 4247 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
6 | 5 | simprbi 271 | . . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
7 | 6 | r19.21bi 2492 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Tr 𝑥) |
8 | ordelss 4259 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴) | |
9 | simpl 108 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴) | |
10 | trssord 4260 | . . . 4 ⊢ ((Tr 𝑥 ∧ 𝑥 ⊆ 𝐴 ∧ Ord 𝐴) → Ord 𝑥) | |
11 | 7, 8, 9, 10 | syl3anc 1197 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) |
12 | 4, 11 | vtoclg 2715 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)) |
13 | 12 | anabsi7 553 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 ∀wral 2388 ⊆ wss 3035 Tr wtr 3984 Ord word 4242 |
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-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-in 3041 df-ss 3048 df-uni 3701 df-tr 3985 df-iord 4246 |
This theorem is referenced by: tron 4262 ordelon 4263 ordsucg 4376 ordwe 4448 smores 6141 |
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