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Mirrors > Home > MPE Home > Th. List > disjdifr | Structured version Visualization version GIF version |
Description: A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.) |
Ref | Expression |
---|---|
disjdifr | ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4217 | . 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∩ 𝐴) | |
2 | disjdif 4478 | . 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
3 | 1, 2 | eqtr3i 2765 | 1 ⊢ ((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∩ cin 3962 ∅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-in 3970 df-ss 3980 df-nul 4340 |
This theorem is referenced by: ssdifin0 4492 fvsnun1 7202 fveqf1o 7322 f1ofvswap 7326 ralxpmap 8935 difsnen 9092 domunsn 9166 limensuci 9192 pssnn 9207 marypha1lem 9471 dif1card 10048 ackbij1lem18 10274 canthp1lem1 10690 grothprim 10872 hashgval 14369 hashun3 14420 hashfun 14473 hashbclem 14488 setsfun 17205 setsfun0 17206 setsid 17242 mreexexlem4d 17692 pwssplit1 21076 islindf4 21876 psdmul 22188 neitr 23204 regsep2 23400 restmetu 24599 volinun 25595 tdeglem4 26114 noetasuplem3 27795 noetasuplem4 27796 difeq 32546 disjdifprg 32595 tocycfvres1 33113 tocycfvres2 33114 cycpmfvlem 33115 cycpmfv3 33118 cycpmcl 33119 rprmdvdsprod 33542 measunl 34197 eulerpartlemt 34353 mthmpps 35567 cldbnd 36309 poimirlem15 37622 poimirlem16 37623 poimirlem19 37626 poimirlem27 37634 selvvvval 42572 evlselvlem 42573 evlselv 42574 eldioph2lem1 42748 eldioph2lem2 42749 diophren 42801 kelac1 43052 isomenndlem 46486 seposep 48722 |
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