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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 4906 | . 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 205 ∧ wa 395 ∈ wcel 2098 ∃wrex 3064 ⊆ wss 3943 ∪ cuni 4902 Oncon0 6357 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rex 3065 df-v 3470 df-uni 4903 |
This theorem is referenced by: (None) |
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