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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 8504 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex1i 5601 | . . 3 ⊢ (𝐴 ≺ ∅ → 𝐴 ∈ V) |
3 | 0domg 8632 | . . 3 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≺ ∅ → ∅ ≼ 𝐴) |
5 | domnsym 8631 | . . 3 ⊢ (∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅) | |
6 | 5 | con2i 141 | . 2 ⊢ (𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴) |
7 | 4, 6 | pm2.65i 195 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 class class class wbr 5057 ≼ cdom 8495 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: domunsn 8655 sdomsdomcardi 9388 canthp1lem1 10062 canthp1lem2 10063 rankcf 10187 |
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