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Theorem ordelon 4356
Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
ordelon ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)

Proof of Theorem ordelon
StepHypRef Expression
1 ordelord 4354 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4346 . . 3 (𝐵𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
32adantl 275 . 2 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On ↔ Ord 𝐵))
41, 3mpbird 166 1 ((Ord 𝐴𝐵𝐴) → 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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