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Mirrors > Home > MPE Home > Th. List > trssord | Structured version Visualization version 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 | wess 5675 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴)) | |
2 | ordwe 6399 | . . . . 5 ⊢ (Ord 𝐵 → E We 𝐵) | |
3 | 1, 2 | impel 505 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴) |
4 | 3 | anim2i 617 | . . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴)) |
5 | 4 | 3impb 1114 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴)) |
6 | df-ord 6389 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
7 | 5, 6 | sylibr 234 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ⊆ wss 3963 Tr wtr 5265 E cep 5588 We wwe 5640 Ord word 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ral 3060 df-ss 3980 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: ordin 6416 ssorduni 7798 ordsuci 7828 sucexeloniOLD 7830 suceloniOLD 7832 ordom 7897 ordtypelem2 9557 hartogs 9582 card2on 9592 tskwe 9988 ondomon 10601 dford3lem2 43016 dford3 43017 iunord 48907 |
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