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Mirrors > Home > MPE Home > Th. List > 0dom | Structured version Visualization version GIF version |
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
0sdom.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
0dom | ⊢ ∅ ≼ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sdom.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | 0domg 9095 | . 2 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∅ ≼ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3466 ∅c0 4314 class class class wbr 5138 ≼ cdom 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-dom 8936 |
This theorem is referenced by: domunsn 9122 mapdom1 9137 mapdom2 9143 fodomfi 9320 marypha1lem 9423 card2inf 9545 iunfictbso 10104 konigthlem 10558 cctop 22819 ovol0 25332 fvconstdomi 47680 |
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