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Theorem onelss 4348
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
onelss (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4336 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 4340 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 114 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 14 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  wss 3102  Ord word 4323  Oncon0 4324
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 3774  df-tr 4064  df-iord 4327  df-on 4329
This theorem is referenced by:  onelssi  4390  ssorduni  4447  onsucelsucr  4468  tfisi  4547  tfrlem9  6267  nntri2or2  6446  phpelm  6812  exmidontri2or  7179  ennnfonelemk  12171
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