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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. (Contributed by NM, 26-Oct-2003.) |
Ref | Expression |
---|---|
ordelon | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordelord 6181 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
2 | elong 6167 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On ↔ Ord 𝐵)) | |
3 | 2 | adantl 485 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On ↔ Ord 𝐵)) |
4 | 1, 3 | mpbird 260 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Ord word 6158 Oncon0 6159 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: onelon 6184 ordunidif 6207 ordpwsuc 7510 ordsucun 7520 ordunel 7522 ordunisuc2 7539 oesuclem 8133 odi 8188 oelim2 8204 oeoalem 8205 oeoelem 8207 limenpsi 8676 ordtypelem9 8974 oismo 8988 cantnflt 9119 cantnfp1lem3 9127 cantnflem1b 9133 cantnflem1 9136 rankr1bg 9216 rankr1clem 9233 rankr1c 9234 rankonidlem 9241 infxpenlem 9424 coflim 9672 fin23lem26 9736 fpwwe2lem8 10048 onsuct0 33902 iunord 45206 |
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