Theorem onuniss2 4525

Description: The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onuniss2	⊢ (𝐴 ∈ On → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem onuniss2

Step	Hyp	Ref	Expression
1		unimax 3857	1 ⊢ (𝐴 ∈ On → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1363   ∈ wcel 2159  {crab 2471   ⊆ wss 3143  ∪ cuni 3823  Oncon0 4377

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rab 2476  df-v 2753  df-in 3149  df-ss 3156  df-uni 3824

This theorem is referenced by: (None)