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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 5532 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 219 | . 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 5559 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 11 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3880 | 1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 ∅c0 4211 × cxp 5523 Rel wrel 5530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-opab 5093 df-xp 5531 df-rel 5532 |
This theorem is referenced by: 0nelrel 5584 reldmtpos 7929 bj-0nelopab 34859 bj-brrelex12ALT 34860 |
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