Theorem onsupuni2 42445

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 4610	. 2 ⊢ (𝐴 ∈ 𝒫 On ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ On))
2		onsupuni 42444	. . 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 216	1 ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ⊆ wss 3948  𝒫 cpw 4602  ∪ cuni 4908   E cep 5579  Oncon0 6364  supcsup 9441

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-iota 6495  df-riota 7368  df-sup 9443

This theorem is referenced by: (None)