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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 9202 | . 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅) | |
2 | sup00 9223 | . 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅ | |
3 | 1, 2 | eqtri 2766 | 1 ⊢ inf(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∅c0 4256 ◡ccnv 5588 supcsup 9199 infcinf 9200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-in 3894 df-ss 3904 df-nul 4257 df-sn 4562 df-uni 4840 df-sup 9201 df-inf 9202 |
This theorem is referenced by: (None) |
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