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Theorem undif3 3758
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 3644 . . . 4
2 pm4.53 492 . . . . 5
3 eldif 3485 . . . . 5
42, 3xchnxbir 309 . . . 4
51, 4anbi12i 697 . . 3
6 eldif 3485 . . 3
7 elun 3644 . . . 4
8 eldif 3485 . . . . 5
98orbi2i 519 . . . 4
10 orc 385 . . . . . . 7
11 olc 384 . . . . . . 7
1210, 11jca 532 . . . . . 6
13 olc 384 . . . . . . 7
14 orc 385 . . . . . . 7
1513, 14anim12i 566 . . . . . 6
1612, 15jaoi 379 . . . . 5
17 simpl 457 . . . . . . 7
1817orcd 392 . . . . . 6
19 olc 384 . . . . . 6
20 orc 385 . . . . . . 7
2120adantr 465 . . . . . 6
2220adantl 466 . . . . . 6
2318, 19, 21, 22ccase 946 . . . . 5
2416, 23impbii 188 . . . 4
257, 9, 243bitri 271 . . 3
265, 6, 253bitr4ri 278 . 2
2726eqriv 2453 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  u.cun 3473
This theorem is referenced by:  undifabs  3905  llycmpkgen2  20051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-un 3480
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