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Theorem oneluni 4139
Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
oneluni (𝐵𝐴 → (𝐴𝐵) = 𝐴)

Proof of Theorem oneluni
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onelssi 4137 . 2 (𝐵𝐴𝐵𝐴)
3 ssequn2 3113 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
42, 3sylib 127 1 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  cun 2912  wss 2914  Oncon0 4071
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-un 2919  df-in 2921  df-ss 2928  df-uni 3577  df-tr 3851  df-iord 4074  df-on 4076
This theorem is referenced by: (None)
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