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Theorem difex2 6588
 Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
difex2

Proof of Theorem difex2
StepHypRef Expression
1 difexg 4581 . 2
2 ssun2 3650 . . . . 5
3 uncom 3630 . . . . . 6
4 undif2 3886 . . . . . 6
53, 4eqtr2i 2471 . . . . 5
62, 5sseqtri 3518 . . . 4
7 unexg 6582 . . . 4
8 ssexg 4579 . . . 4
96, 7, 8sylancr 663 . . 3
109expcom 435 . 2
111, 10impbid2 204 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wcel 1802  cvv 3093   cdif 3455   cun 3456   wss 3458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-sn 4011  df-pr 4013  df-uni 4231 This theorem is referenced by:  elpwun  6594
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