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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 4400 | . 2 ⊢ ∅ ⊆ (V × V) | |
| 2 | df-rel 5692 | . 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V)) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ Rel ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3480 ⊆ wss 3951 ∅c0 4333 × cxp 5683 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-dif 3954 df-ss 3968 df-nul 4334 df-rel 5692 | 
| This theorem is referenced by: relsnb 5812 reldm0 5938 cnveq0 6217 co02 6280 co01 6281 tpos0 8281 0we1 8544 0er 8783 canthwe 10691 relexpreld 15079 disjALTV0 38755 dibvalrel 41165 dicvalrelN 41187 dihvalrel 41281 | 
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