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| 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 4136 | . 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅ | |
| 2 | ss0 4402 | . 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (∅ ∖ 𝐴) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 |
| This theorem is referenced by: symdif0 5085 fresaun 6779 dffv2 7004 ablfac1eulem 20092 itgioo 25851 newval 27894 imadifxp 32614 sibf0 34336 ballotlemfval0 34498 ballotlemgun 34527 satf0 35377 mdvval 35509 fzdifsuc2 45322 ibliooicc 45986 disjdifb 48729 |
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