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Theorem oneluni 4313
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 4311 . 2 (𝐵𝐴𝐵𝐴)
3 ssequn2 3215 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
42, 3sylib 121 1 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  cun 3035  wss 3037  Oncon0 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-uni 3703  df-tr 3987  df-iord 4248  df-on 4250
This theorem is referenced by: (None)
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