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Mirrors > Home > ILE Home > Th. List > ordelon | 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 4354 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | |
2 | elong 4346 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On ↔ Ord 𝐵)) | |
3 | 2 | adantl 275 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On ↔ Ord 𝐵)) |
4 | 1, 3 | mpbird 166 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 Ord word 4335 Oncon0 4336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-in 3118 df-ss 3125 df-uni 3785 df-tr 4076 df-iord 4339 df-on 4341 |
This theorem is referenced by: onelon 4357 ordsson 4464 ordpwsucss 4539 tfr1onlemsucfn 6300 tfr1onlemsucaccv 6301 tfr1onlembfn 6304 tfr1onlemubacc 6306 tfr1onlemaccex 6308 tfrcllemsucfn 6313 tfrcllemsucaccv 6314 tfrcllembfn 6317 tfrcllemubacc 6319 tfrcllemaccex 6321 tfrcl 6324 |
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