Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  undif3 Unicode version

Theorem undif3 3336
 Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.)
Assertion
Ref Expression
undif3

Proof of Theorem undif3
StepHypRef Expression
1 elun 3226 . . . 4
2 pm4.53 480 . . . . 5
3 eldif 3088 . . . . 5
42, 3xchnxbir 302 . . . 4
51, 4anbi12i 681 . . 3
6 eldif 3088 . . 3
7 elun 3226 . . . 4
8 eldif 3088 . . . . 5
98orbi2i 507 . . . 4
10 orc 376 . . . . . . 7
11 olc 375 . . . . . . 7
1210, 11jca 520 . . . . . 6
13 olc 375 . . . . . . 7
14 orc 376 . . . . . . 7
1513, 14anim12i 551 . . . . . 6
1612, 15jaoi 370 . . . . 5
17 simpl 445 . . . . . . 7
1817orcd 383 . . . . . 6
19 olc 375 . . . . . 6
20 orc 376 . . . . . . 7
2120adantr 453 . . . . . 6
2220adantl 454 . . . . . 6
2318, 19, 21, 22ccase 917 . . . . 5
2416, 23impbii 182 . . . 4
257, 9, 243bitri 264 . . 3
265, 6, 253bitr4ri 271 . 2
2726eqriv 2250 1
 Colors of variables: wff set class Syntax hints:   wn 5   wo 359   wa 360   wceq 1619   wcel 1621   cdif 3075   cun 3076 This theorem is referenced by:  undifabs  3437  llycmpkgen2  17077 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-un 3083
 Copyright terms: Public domain W3C validator