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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 4022 | . 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅ | |
2 | ss0 4287 | . 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∅ ∖ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3840 ⊆ wss 3843 ∅c0 4211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-nul 4212 |
This theorem is referenced by: symdif0 4970 fresaun 6549 dffv2 6763 ablfac1eulem 19313 itgioo 24568 nbgr0vtx 27298 imadifxp 30514 sibf0 31871 ballotlemfval0 32032 ballotlemgun 32061 satf0 32905 mdvval 33037 fzdifsuc2 42387 ibliooicc 43054 disjdifb 45687 |
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