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Theorem undif3 3604
 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 3490 . . . 4
2 pm4.53 480 . . . . 5
3 eldif 3332 . . . . 5
42, 3xchnxbir 302 . . . 4
51, 4anbi12i 680 . . 3
6 eldif 3332 . . 3
7 elun 3490 . . . 4
8 eldif 3332 . . . . 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 914 . . . . 5
2416, 23impbii 182 . . . 4
257, 9, 243bitri 264 . . 3
265, 6, 253bitr4ri 271 . 2
2726eqriv 2435 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 359   wa 360   wceq 1653   wcel 1726   cdif 3319   cun 3320 This theorem is referenced by:  undifabs  3707  llycmpkgen2  17584 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-un 3327
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