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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 2100 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | anbi2d 437 | . . . 4 ⊢ (𝑥 = 𝐵 → ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (Ord 𝐴 ∧ 𝐵 ∈ 𝐴))) |
3 | ordeq 4109 | . . . 4 ⊢ (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵)) | |
4 | 2, 3 | imbi12d 223 | . . 3 ⊢ (𝑥 = 𝐵 → (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) ↔ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵))) |
5 | dford3 4104 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
6 | 5 | simprbi 260 | . . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
7 | 6 | r19.21bi 2407 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Tr 𝑥) |
8 | ordelss 4116 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴) | |
9 | simpl 102 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴) | |
10 | trssord 4117 | . . . 4 ⊢ ((Tr 𝑥 ∧ 𝑥 ⊆ 𝐴 ∧ Ord 𝐴) → Ord 𝑥) | |
11 | 7, 8, 9, 10 | syl3anc 1135 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) |
12 | 4, 11 | vtoclg 2613 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)) |
13 | 12 | anabsi7 515 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ⊆ wss 2917 Tr wtr 3854 Ord word 4099 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 |
This theorem is referenced by: tron 4119 ordelon 4120 ordsucg 4228 ordwe 4300 smores 5907 |
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