Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0sdom | Structured version Visualization version GIF version |
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.) |
Ref | Expression |
---|---|
0sdom.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
0sdom | ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sdom.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | 0sdomg 8640 | . 2 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 class class class wbr 5058 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 |
This theorem is referenced by: sdom1 8712 marypha1lem 8891 konigthlem 9984 pwcfsdom 9999 cfpwsdom 10000 rankcf 10193 r1tskina 10198 1stcfb 22047 snct 30443 sigapildsys 31416 |
Copyright terms: Public domain | W3C validator |