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Theorem onelssi 4388
 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 4346 . 2 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
31, 2ax-mp 5 1 (𝐵𝐴𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 2128   ⊆ wss 3102  Oncon0 4322 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063  df-iord 4325  df-on 4327 This theorem is referenced by:  onelini  4389  oneluni  4390  omp1eomlem  7028  enumctlemm  7048  ennnfonelemdc  12100  ennnfonelemg  12104  ctinfom  12129  2o01f  13528  isomninnlem  13563  iswomninnlem  13582  ismkvnnlem  13585
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