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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 4376 | . . 3 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
| 2 | 1 | difeq2i 4123 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = (𝐴 ∖ ∅) |
| 3 | difdif 4135 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = 𝐴 | |
| 4 | 2, 3 | eqtr3i 2767 | 1 ⊢ (𝐴 ∖ ∅) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∅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-rab 3437 df-v 3482 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: unvdif 4475 disjdif2 4480 csbdif 4524 iinvdif 5080 symdif0 5085 dffv2 7004 2oconcl 8541 oe0m0 8558 oev2 8561 infdiffi 9698 cnfcom2lem 9741 brttrcl2 9754 ttrcltr 9756 rnttrcl 9762 m1bits 16477 mreexdomd 17692 efgi0 19738 vrgpinv 19787 frgpuptinv 19789 frgpnabllem1 19891 gsumval3 19925 gsumcllem 19926 dprddisj2 20059 0cld 23046 indiscld 23099 mretopd 23100 hauscmplem 23414 cfinfil 23901 csdfil 23902 filufint 23928 bcth3 25365 rembl 25575 volsup 25591 new0 27913 disjdifprg 32588 tocycf 33137 tocyc01 33138 prsiga 34132 sigapildsyslem 34162 sigapildsys 34163 sxbrsigalem3 34274 0elcarsg 34309 carsgclctunlem3 34322 onint1 36450 lindsdom 37621 oe0rif 43298 tfsconcat0i 43358 ntrclscls00 44079 ntrclskb 44082 compne 44460 prsal 46333 saluni 46340 caragen0 46521 carageniuncllem1 46536 iscnrm3rlem4 48840 aacllem 49320 |
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