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Theorem oneluni 4430
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 4428 . 2 (𝐵𝐴𝐵𝐴)
3 ssequn2 3308 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
42, 3sylib 122 1 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cun 3127  wss 3129  Oncon0 4362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367
This theorem is referenced by: (None)
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