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Mirrors > Home > MPE Home > Th. List > 0nelfun | Structured version Visualization version GIF version |
Description: A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
Ref | Expression |
---|---|
0nelfun | ⊢ (Fun 𝑅 → ∅ ∉ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6576 | . 2 ⊢ (Fun 𝑅 → Rel 𝑅) | |
2 | 0nelrel 5743 | . 2 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Fun 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∉ wnel 3036 ∅c0 4325 Rel wrel 5687 Fun wfun 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-nel 3037 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-opab 5216 df-xp 5688 df-rel 5689 df-fun 6556 |
This theorem is referenced by: (None) |
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