Theorem onuniss2 4489

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 3823	1 ⊢ (𝐴 ∈ On → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  {crab 2448   ⊆ wss 3116  ∪ cuni 3789  Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790

This theorem is referenced by: (None)