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Theorem onelss 4152
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 4140 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 4144 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 112 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 14 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  wss 2945  Ord word 4127  Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by:  onelssi  4194  ssorduni  4241  onsucelsucr  4262  tfisi  4338  tfrlem9  5966  nntri2or2  6107  phpelm  6359
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