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Mirrors > Home > MPE Home > Th. List > rel0 | Structured version Visualization version GIF version |
Description: The empty set is a relation. (Contributed by NM, 26-Apr-1998.) |
Ref | Expression |
---|---|
rel0 | ⊢ Rel ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4394 | . 2 ⊢ ∅ ⊆ (V × V) | |
2 | df-rel 5681 | . 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V)) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ Rel ∅ |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3475 ⊆ wss 3946 ∅c0 4320 × cxp 5672 Rel wrel 5679 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3949 df-in 3953 df-ss 3963 df-nul 4321 df-rel 5681 |
This theorem is referenced by: relsnb 5799 reldm0 5924 cnveq0 6192 co02 6255 co01 6256 tpos0 8235 0we1 8500 0er 8735 canthwe 10641 relexpreld 14982 disjALTV0 37561 dibvalrel 39971 dicvalrelN 39993 dihvalrel 40087 |
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