Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > undif | Structured version Visualization version GIF version |
Description: Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
undif | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 3745 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
2 | undif2 3996 | . . 3 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
3 | 2 | eqeq1i 2615 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: raldifeq 4011 difsnid 4282 fveqf1o 6457 ralxpmap 7793 undifixp 7830 dfdom2 7867 sbthlem5 7959 sbthlem6 7960 domunsn 7995 fodomr 7996 mapdom2 8016 limensuci 8021 findcard2 8085 unfi 8112 marypha1lem 8222 brwdom2 8361 infdifsn 8437 ackbij1lem12 8936 ackbij1lem18 8942 ssfin4 9015 fin23lem28 9045 fin23lem30 9047 fin1a2lem13 9117 canthp1lem1 9353 xrsupss 12011 xrinfmss 12012 hashssdif 13061 hashfun 13084 hashf1lem2 13097 fsumless 14369 incexclem 14407 incexc 14408 fprodsplit1f 14560 pwssplit1 18880 frlmsslss2 19933 mdetdiaglem 20223 mdetrlin 20227 mdetrsca 20228 mdetralt 20233 smadiadet 20295 isclo 20701 cmpcld 21015 rrxcph 22988 rrxdstprj1 23000 uniiccmbl 23164 itgss3 23387 dchreq 24783 axlowdimlem7 25628 axlowdimlem10 25631 padct 28885 resf1o 28893 locfinref 29236 indval2 29404 esummono 29443 gsumesum 29448 sigaclfu2 29511 measxun2 29600 measvuni 29604 measssd 29605 pmeasmono 29713 eulerpartlemt 29760 poimirlem9 32588 poimirlem15 32594 poimirlem25 32604 diophrw 36340 eldioph2lem1 36341 eldioph2lem2 36342 kelac1 36651 fsumsplit1 38639 ioccncflimc 38771 icocncflimc 38775 dirkercncflem2 38997 dirkercncflem3 38998 sge0ss 39305 meassle 39356 meadif 39372 meaiininclem 39376 isomenndlem 39420 hspmbllem1 39516 hspmbllem2 39517 ovolval4lem1 39539 fsumsplitsndif 40372 |
Copyright terms: Public domain | W3C validator |