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Theorem ordelon 4210
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 4208 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4200 . . 3 (𝐵𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
32adantl 271 . 2 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On ↔ Ord 𝐵))
41, 3mpbird 165 1 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1438  Ord word 4189  Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  onelon  4211  ordsson  4309  ordpwsucss  4383  tfr1onlemsucfn  6105  tfr1onlemsucaccv  6106  tfr1onlembfn  6109  tfr1onlemubacc  6111  tfr1onlemaccex  6113  tfrcllemsucfn  6118  tfrcllemsucaccv  6119  tfrcllembfn  6122  tfrcllemubacc  6124  tfrcllemaccex  6126  tfrcl  6129
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