Theorem onsupuni2 44175

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 4564	. 2 ⊢ (𝐴 ∈ 𝒫 On ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ On))
2		onsupuni 44174	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ V) → sup(𝐴, On, E ) = ∪ 𝐴)
3	2	ancoms 464	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ On) → sup(𝐴, On, E ) = ∪ 𝐴)
4	1, 3	sylbi 220	1 ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3898  𝒫 cpw 4556  ∪ cuni 4866   E cep 5546  Oncon0 6351  supcsup 9410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-iota 6483  df-riota 7365  df-sup 9412

This theorem is used by: (None)