Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0dif | Structured version Visualization version GIF version |
Description: The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
0dif | ⊢ (∅ ∖ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4110 | . 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅ | |
2 | ss0 4354 | . 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∅ ∖ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 |
This theorem is referenced by: symdif0 5009 fresaun 6551 dffv2 6758 ablfac1eulem 19196 itgioo 24418 nbgr0vtx 27140 imadifxp 30353 sibf0 31594 ballotlemfval0 31755 ballotlemgun 31784 satf0 32621 mdvval 32753 fzdifsuc2 41584 ibliooicc 42263 |
Copyright terms: Public domain | W3C validator |