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Theorem onelss 4096
 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 4084 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 4088 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 108 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 14 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ wss 2914  Ord word 4071  Oncon0 4072 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 2308  df-rex 2309  df-v 2556  df-in 2921  df-ss 2928  df-uni 3578  df-tr 3852  df-iord 4075  df-on 4077 This theorem is referenced by:  onelssi  4138  ssorduni  4185  onsucelsucr  4206  tfisi  4273  tfrlem9  5898  nntri2or2  6039  phpelm  6291
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