![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5299 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴)) | |
2 | ordwe 5954 | . . . . 5 ⊢ (Ord 𝐵 → E We 𝐵) | |
3 | 1, 2 | impel 502 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴) |
4 | 3 | anim2i 611 | . . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴)) |
5 | 4 | 3impb 1144 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴)) |
6 | df-ord 5944 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
7 | 5, 6 | sylibr 226 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 ⊆ wss 3769 Tr wtr 4945 E cep 5224 We wwe 5270 Ord word 5940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-ral 3094 df-in 3776 df-ss 3783 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 |
This theorem is referenced by: ordin 5971 ssorduni 7219 suceloni 7247 ordom 7308 ordtypelem2 8666 hartogs 8691 card2on 8701 tskwe 9062 ondomon 9673 dford3lem2 38379 dford3 38380 iunord 43221 |
Copyright terms: Public domain | W3C validator |