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Theorem onsupintrab2 42721
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 42720 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ V) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
32ancoms 457 . 2 ((𝐴 ∈ V ∧ 𝐴 ⊆ On) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
41, 3sylbi 216 1 (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  {crab 3419  Vcvv 3463  wss 3941  𝒫 cpw 4599   cint 4945   E cep 5576  Oncon0 6365  supcsup 9458
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 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-br 5145  df-opab 5207  df-tr 5262  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-ord 6368  df-on 6369  df-iota 6495  df-riota 7369  df-sup 9460
This theorem is referenced by: (None)
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