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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 6215 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
2 | elong 6201 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On ↔ Ord 𝐵)) | |
3 | 2 | adantl 484 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On ↔ Ord 𝐵)) |
4 | 1, 3 | mpbird 259 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Ord word 6192 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: onelon 6218 ordunidif 6241 ordpwsuc 7532 ordsucun 7542 ordunel 7544 ordunisuc2 7561 oesuclem 8152 odi 8207 oelim2 8223 oeoalem 8224 oeoelem 8226 limenpsi 8694 ordtypelem9 8992 oismo 9006 cantnflt 9137 cantnfp1lem3 9145 cantnflem1b 9151 cantnflem1 9154 rankr1bg 9234 rankr1clem 9251 rankr1c 9252 rankonidlem 9259 infxpenlem 9441 coflim 9685 fin23lem26 9749 fpwwe2lem8 10061 onsuct0 33791 iunord 44786 |
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