| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > br0 | Structured version Visualization version GIF version | ||
| Description: The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.) |
| Ref | Expression |
|---|---|
| br0 | ⊢ ¬ 𝐴∅𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . 2 ⊢ ¬ 〈𝐴, 𝐵〉 ∈ ∅ | |
| 2 | df-br 5144 | . 2 ⊢ (𝐴∅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ∅) | |
| 3 | 1, 2 | mtbir 323 | 1 ⊢ ¬ 𝐴∅𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ∅c0 4333 〈cop 4632 class class class wbr 5143 |
| 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-nul 4334 df-br 5144 |
| This theorem is referenced by: sbcbr123 5197 sbcbr 5198 cnv0 6160 co02 6280 fvmptopabOLD 7488 brfvopab 7490 0we1 8544 brdom3 10568 canthwe 10691 relexpindlem 15102 join0 18450 meet0 18451 acycgr0v 35153 prclisacycgr 35156 disjALTV0 38755 brnonrel 43602 upwlkbprop 48054 |
| Copyright terms: Public domain | W3C validator |