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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 5506 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴)) | |
2 | ordwe 6172 | . . . . 5 ⊢ (Ord 𝐵 → E We 𝐵) | |
3 | 1, 2 | impel 509 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴) |
4 | 3 | anim2i 619 | . . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴)) |
5 | 4 | 3impb 1112 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴)) |
6 | df-ord 6162 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
7 | 5, 6 | sylibr 237 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ⊆ wss 3881 Tr wtr 5136 E cep 5429 We wwe 5477 Ord word 6158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 |
This theorem is referenced by: ordin 6189 ssorduni 7480 suceloni 7508 ordom 7569 ordtypelem2 8967 hartogs 8992 card2on 9002 tskwe 9363 ondomon 9974 dford3lem2 39968 dford3 39969 iunord 45206 |
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