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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 5692 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) | |
2 | df-nel 3050 | . 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∉ wnel 3049 ∅c0 4282 Rel wrel 5638 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-nel 3050 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-opab 5168 df-xp 5639 df-rel 5640 |
This theorem is referenced by: 0nelfun 6519 |
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