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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsuplub | Structured version Visualization version GIF version |
Description: The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025.) |
Ref | Expression |
---|---|
onsuplub | ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 4935 | . 2 ⊢ (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧) | |
2 | 1 | a1i 11 | 1 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2103 ∃wrex 3072 ⊆ wss 3970 ∪ cuni 4931 Oncon0 6394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3073 df-v 3484 df-uni 4932 |
This theorem is referenced by: (None) |
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