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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 5253 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴)) | |
2 | ordwe 5897 | . . . . 5 ⊢ (Ord 𝐵 → E We 𝐵) | |
3 | 1, 2 | impel 486 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴) |
4 | 3 | anim2i 594 | . . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴)) |
5 | 4 | 3impb 1108 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴)) |
6 | df-ord 5887 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
7 | 5, 6 | sylibr 224 | 1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ⊆ wss 3715 Tr wtr 4904 E cep 5178 We wwe 5224 Ord word 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-ral 3055 df-in 3722 df-ss 3729 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 |
This theorem is referenced by: ordin 5914 ssorduni 7151 suceloni 7179 ordom 7240 ordtypelem2 8591 hartogs 8616 card2on 8626 tskwe 8986 ondomon 9597 dford3lem2 38114 dford3 38115 iunord 42950 |
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