Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oneluni Structured version   Visualization version   GIF version

Theorem oneluni 5757
 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 5753 . 2 (𝐵𝐴𝐵𝐴)
3 ssequn2 3748 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
42, 3sylib 207 1 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  onun2i  5760
 Copyright terms: Public domain W3C validator