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Mirrors > Home > ILE Home > Th. List > br0 | 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 3450 | . 2 ⊢ ¬ 〈𝐴, 𝐵〉 ∈ ∅ | |
2 | df-br 4030 | . 2 ⊢ (𝐴∅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ∅) | |
3 | 1, 2 | mtbir 672 | 1 ⊢ ¬ 𝐴∅𝐵 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2164 ∅c0 3446 〈cop 3621 class class class wbr 4029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-dif 3155 df-nul 3447 df-br 4030 |
This theorem is referenced by: (None) |
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