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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 8637 | . 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅) | |
2 | sup00 8658 | . 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅ | |
3 | 1, 2 | eqtri 2802 | 1 ⊢ inf(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∅c0 4141 ◡ccnv 5354 supcsup 8634 infcinf 8635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-in 3799 df-ss 3806 df-nul 4142 df-sn 4399 df-uni 4672 df-sup 8636 df-inf 8637 |
This theorem is referenced by: (None) |
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