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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 5365, ax-un 7740. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
sdom0 | ⊢ ¬ 𝐴 ≺ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom0 9127 | . . . 4 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) | |
2 | en0 9038 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
3 | 1, 2 | sylbb2 237 | . . 3 ⊢ (𝐴 ≼ ∅ → 𝐴 ≈ ∅) |
4 | iman 401 | . . 3 ⊢ ((𝐴 ≼ ∅ → 𝐴 ≈ ∅) ↔ ¬ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅)) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ ¬ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅) |
6 | brsdom 8996 | . 2 ⊢ (𝐴 ≺ ∅ ↔ (𝐴 ≼ ∅ ∧ ¬ 𝐴 ≈ ∅)) | |
7 | 5, 6 | mtbir 323 | 1 ⊢ ¬ 𝐴 ≺ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∅c0 4323 class class class wbr 5148 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-en 8965 df-dom 8966 df-sdom 8967 |
This theorem is referenced by: domunsn 9152 sdomsdomcardi 9995 canthp1lem1 10676 canthp1lem2 10677 rankcf 10801 |
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