Theorem onsupuni2 42723

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 4606	. 2 ⊢ (𝐴 ∈ 𝒫 On ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ On))
2		onsupuni 42722	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ V) → sup(𝐴, On, E ) = ∪ 𝐴)
3	2	ancoms 457	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ On) → sup(𝐴, On, E ) = ∪ 𝐴)
4	1, 3	sylbi 216	1 ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ⊆ wss 3939  𝒫 cpw 4598  ∪ cuni 4903   E cep 5575  Oncon0 6364  supcsup 9463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-iota 6495  df-riota 7372  df-sup 9465

This theorem is referenced by: (None)