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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 5562 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 218 | . 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 5589 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 11 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3970 | 1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 × cxp 5553 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: 0nelrel 5613 reldmtpos 7900 bj-0nelopab 34361 bj-brrelex12ALT 34362 |
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