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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 6408 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
2 | elong 6394 | . . 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 2106 Ord word 6385 Oncon0 6386 |
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 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: onelon 6411 ordunidif 6435 ordpwsuc 7835 ordsucun 7845 ordunel 7847 ordunisuc2 7865 oesuclem 8562 odi 8616 oelim2 8632 oeoalem 8633 oeoelem 8635 limenpsi 9191 ordtypelem9 9564 oismo 9578 cantnflt 9710 cantnfp1lem3 9718 cantnflem1b 9724 cantnflem1 9727 rankr1bg 9841 rankr1clem 9858 rankr1c 9859 rankonidlem 9866 infxpenlem 10051 coflim 10299 fin23lem26 10363 fpwwe2lem7 10675 onsuct0 36424 ordnexbtwnsuc 43257 orddif0suc 43258 omord2lim 43290 nadd2rabtr 43374 nadd2rabex 43376 nadd1rabtr 43378 nadd1rabex 43380 iunord 48907 |
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