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Mirrors > Home > MPE Home > Th. List > sdom0 | Structured version Visualization version GIF version |
Description: The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) |
Ref | Expression |
---|---|
sdom0 | ⊢ ¬ 𝐴 ≺ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8248 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5406 | . . 3 ⊢ (𝐴 ≺ ∅ → 𝐴 ∈ V) |
3 | 0domg 8375 | . . 3 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≺ ∅ → ∅ ≼ 𝐴) |
5 | domnsym 8374 | . . 3 ⊢ (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅) | |
6 | 5 | con2i 137 | . 2 ⊢ (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴) |
7 | 4, 6 | pm2.65i 186 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 class class class wbr 4886 ≼ cdom 8239 ≺ csdm 8240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 |
This theorem is referenced by: domunsn 8398 sdomsdomcardi 9130 canthp1lem1 9809 canthp1lem2 9810 rankcf 9934 |
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