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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 6341 | . 2 ⊢ (Fun 𝑅 → Rel 𝑅) | |
2 | 0nelrel 5577 | . 2 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Fun 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∉ wnel 3091 ∅c0 4243 Rel wrel 5524 Fun wfun 6318 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-fun 6326 |
This theorem is referenced by: (None) |
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