![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0domg | 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 |
---|---|
0domg | ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4005 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssdomg 8043 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ⊆ 𝐴 → ∅ ≼ 𝐴)) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 ⊆ wss 3607 ∅c0 3948 class class class wbr 4685 ≼ cdom 7995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-dom 7999 |
This theorem is referenced by: dom0 8129 0sdomg 8130 0dom 8131 sdom0 8133 carddomi2 8834 wdomfil 8922 wdomnumr 8925 hashge0 13214 ufildom1 21777 harn0 37989 |
Copyright terms: Public domain | W3C validator |