Theorem onsupuni2 43333

Description: The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.)

Assertion

Ref	Expression
onsupuni2	⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

Proof of Theorem onsupuni2

Step	Hyp	Ref	Expression
1		elpwb 4555	. 2 ⊢ (𝐴 ∈ 𝒫 On ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ On))
2		onsupuni 43332	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ V) → sup(𝐴, On, E ) = ∪ 𝐴)
3	2	ancoms 458	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ On) → sup(𝐴, On, E ) = ∪ 𝐴)
4	1, 3	sylbi 217	1 ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856   E cep 5513  Oncon0 6306  supcsup 9324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310  df-iota 6437  df-riota 7303  df-sup 9326

This theorem is referenced by: (None)