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Mirrors > Home > MPE Home > Th. List > 0nelrel0 | Structured version Visualization version GIF version |
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) (Revised by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
0nelrel0 | ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5638 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 215 | . 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 5665 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 11 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3945 | 1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 × cxp 5629 Rel wrel 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-opab 5166 df-xp 5637 df-rel 5638 |
This theorem is referenced by: 0nelrel 5691 reldmtpos 8157 bj-0nelopab 35475 bj-brrelex12ALT 35476 |
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