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| Mirrors > Home > MPE Home > Th. List > ordelon | Structured version Visualization version GIF version | ||
| Description: An element of an ordinal class is an ordinal number. Lemma 1.3 of [Schloeder] p. 1. (Contributed by NM, 26-Oct-2003.) |
| Ref | Expression |
|---|---|
| ordelon | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordelord 6406 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
| 2 | elong 6392 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On ↔ Ord 𝐵)) | |
| 3 | 2 | adantl 481 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On ↔ Ord 𝐵)) |
| 4 | 1, 3 | mpbird 257 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: onelon 6409 ordunidif 6433 ordpwsuc 7835 ordsucun 7845 ordunel 7847 ordunisuc2 7865 oesuclem 8563 odi 8617 oelim2 8633 oeoalem 8634 oeoelem 8636 limenpsi 9192 ordtypelem9 9566 oismo 9580 cantnflt 9712 cantnfp1lem3 9720 cantnflem1b 9726 cantnflem1 9729 rankr1bg 9843 rankr1clem 9860 rankr1c 9861 rankonidlem 9868 infxpenlem 10053 coflim 10301 fin23lem26 10365 fpwwe2lem7 10677 onsuct0 36442 ordnexbtwnsuc 43280 orddif0suc 43281 omord2lim 43313 nadd2rabtr 43397 nadd2rabex 43399 nadd1rabtr 43401 nadd1rabex 43403 iunord 49195 |
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