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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsupuni2 | Structured version Visualization version GIF version |
Description: The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.) |
Ref | Expression |
---|---|
onsupuni2 | ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴) |
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) |
Colors of variables: wff setvar class |
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) |
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