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Theorem onelssi 4166
 Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onelssi (𝐵𝐴𝐵𝐴)

Proof of Theorem onelssi
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 onelss 4124 . 2 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
31, 2ax-mp 7 1 (𝐵𝐴𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ wss 2917  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  onelini  4167  oneluni  4168
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