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Theorem onsupintrab2 43249
Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
onsupintrab2 (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem onsupintrab2
StepHypRef Expression
1 elpwb 4607 . 2 (𝐴 ∈ 𝒫 On ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ On))
2 onsupintrab 43248 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ V) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
32ancoms 458 . 2 ((𝐴 ∈ V ∧ 𝐴 ⊆ On) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
41, 3sylbi 217 1 (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   cint 4945   E cep 5582  Oncon0 6383  supcsup 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-iota 6513  df-riota 7389  df-sup 9483
This theorem is referenced by: (None)
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