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Mirrors > Home > ILE Home > Th. List > trssord | GIF version |
Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
Ref | Expression |
---|---|
trssord | ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dford3 4289 | . . . . . . 7 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥)) | |
2 | 1 | simprbi 273 | . . . . . 6 ⊢ (Ord 𝐵 → ∀𝑥 ∈ 𝐵 Tr 𝑥) |
3 | ssralv 3161 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 Tr 𝑥 → ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
4 | 2, 3 | syl5 32 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝑥)) |
5 | 4 | imp 123 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → ∀𝑥 ∈ 𝐴 Tr 𝑥) |
6 | 5 | anim2i 339 | . . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) |
7 | 6 | 3impb 1177 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) |
8 | dford3 4289 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
9 | 7, 8 | sylibr 133 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∀wral 2416 ⊆ wss 3071 Tr wtr 4026 Ord word 4284 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-ral 2421 df-in 3077 df-ss 3084 df-iord 4288 |
This theorem is referenced by: ordelord 4303 ordin 4307 ssorduni 4403 ordtriexmidlem 4435 ordtri2or2exmidlem 4441 onsucelsucexmidlem 4444 ordsuc 4478 |
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