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Theorem undif3 3515
 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
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . . 4
2 pm4.53 478 . . . . 5
3 eldif 3221 . . . . 5
42, 3xchnxbir 300 . . . 4
51, 4anbi12i 678 . . 3
6 eldif 3221 . . 3
7 elun 3220 . . . 4
8 eldif 3221 . . . . 5
98orbi2i 505 . . . 4
10 orc 374 . . . . . . 7
11 olc 373 . . . . . . 7
1210, 11jca 518 . . . . . 6
13 olc 373 . . . . . . 7
14 orc 374 . . . . . . 7
1513, 14anim12i 549 . . . . . 6
1612, 15jaoi 368 . . . . 5
17 simpl 443 . . . . . . 7
1817orcd 381 . . . . . 6
19 olc 373 . . . . . 6
20 orc 374 . . . . . . 7
2120adantr 451 . . . . . 6
2220adantl 452 . . . . . 6
2318, 19, 21, 22ccase 912 . . . . 5
2416, 23impbii 180 . . . 4
257, 9, 243bitri 262 . . 3
265, 6, 253bitr4ri 269 . 2
2726eqriv 2350 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 357   wa 358   wceq 1642   wcel 1710   cdif 3206   cun 3207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215 This theorem is referenced by:  undifabs  3627
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