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Mirrors > Home > MPE Home > Th. List > dif0 | Structured version Visualization version GIF version |
Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
dif0 | ⊢ (𝐴 ∖ ∅) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difid 4382 | . . 3 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
2 | 1 | difeq2i 4133 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = (𝐴 ∖ ∅) |
3 | difdif 4145 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = 𝐴 | |
4 | 2, 3 | eqtr3i 2765 | 1 ⊢ (𝐴 ∖ ∅) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∅c0 4339 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-nul 4340 |
This theorem is referenced by: unvdif 4481 disjdif2 4486 csbdif 4530 iinvdif 5085 symdif0 5090 dffv2 7004 2oconcl 8540 oe0m0 8557 oev2 8560 infdiffi 9696 cnfcom2lem 9739 brttrcl2 9752 ttrcltr 9754 rnttrcl 9760 m1bits 16474 mreexdomd 17694 efgi0 19753 vrgpinv 19802 frgpuptinv 19804 frgpnabllem1 19906 gsumval3 19940 gsumcllem 19941 dprddisj2 20074 0cld 23062 indiscld 23115 mretopd 23116 hauscmplem 23430 cfinfil 23917 csdfil 23918 filufint 23944 bcth3 25379 rembl 25589 volsup 25605 new0 27928 disjdifprg 32595 tocycf 33120 tocyc01 33121 prsiga 34112 sigapildsyslem 34142 sigapildsys 34143 sxbrsigalem3 34254 0elcarsg 34289 carsgclctunlem3 34302 onint1 36432 lindsdom 37601 oe0rif 43275 tfsconcat0i 43335 ntrclscls00 44056 ntrclskb 44059 compne 44437 prsal 46274 saluni 46281 caragen0 46462 carageniuncllem1 46477 iscnrm3rlem4 48740 aacllem 49032 |
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