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Mirrors > Home > MPE Home > Th. List > 0nelrel | Structured version Visualization version GIF version |
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
Ref | Expression |
---|---|
0nelrel | ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelrel0 5593 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) | |
2 | df-nel 3040 | . 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅) | |
3 | 1, 2 | sylibr 237 | 1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 ∉ wnel 3039 ∅c0 4221 Rel wrel 5540 |
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 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
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 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-nel 3040 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-opab 5103 df-xp 5541 df-rel 5542 |
This theorem is referenced by: 0nelfun 6368 |
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