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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.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
sdom0 | ⊢ ¬ 𝐴 ≺ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom0 9053 | . . . 4 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) | |
2 | en0 8964 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
3 | 1, 2 | sylbb2 237 | . . 3 ⊢ (𝐴 ≼ ∅ → 𝐴 ≈ ∅) |
4 | iman 403 | . . 3 ⊢ ((𝐴 ≼ ∅ → 𝐴 ≈ ∅) ↔ ¬ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅)) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ ¬ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅) |
6 | brsdom 8922 | . 2 ⊢ (𝐴 ≺ ∅ ↔ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅)) | |
7 | 5, 6 | mtbir 323 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∅c0 4287 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: domunsn 9078 sdomsdomcardi 9914 canthp1lem1 10595 canthp1lem2 10596 rankcf 10720 |
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