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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 6235 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
2 | elong 6221 | . . 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 2110 Ord word 6212 Oncon0 6213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-ord 6216 df-on 6217 |
This theorem is referenced by: onelon 6238 ordunidif 6261 ordpwsuc 7594 ordsucun 7604 ordunel 7606 ordunisuc2 7623 oesuclem 8252 odi 8307 oelim2 8323 oeoalem 8324 oeoelem 8326 limenpsi 8821 ordtypelem9 9142 oismo 9156 cantnflt 9287 cantnfp1lem3 9295 cantnflem1b 9301 cantnflem1 9304 rankr1bg 9419 rankr1clem 9436 rankr1c 9437 rankonidlem 9444 infxpenlem 9627 coflim 9875 fin23lem26 9939 fpwwe2lem7 10251 onsuct0 34367 iunord 46053 |
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