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Mirrors > Home > MPE Home > Th. List > inf00 | Structured version Visualization version GIF version |
Description: The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
inf00 | ⊢ inf(𝐵, ∅, 𝑅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 9474 | . 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅) | |
2 | sup00 9495 | . 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅ | |
3 | 1, 2 | eqtri 2756 | 1 ⊢ inf(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∅c0 4326 ◡ccnv 5681 supcsup 9471 infcinf 9472 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4327 df-sn 4633 df-uni 4913 df-sup 9473 df-inf 9474 |
This theorem is referenced by: (None) |
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