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Theorem difex2 6516
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 4557 . 2
2 ssun2 3634 . . . . 5
3 uncom 3614 . . . . . 6
4 undif2 3869 . . . . . 6
53, 4eqtr2i 2484 . . . . 5
62, 5sseqtri 3502 . . . 4
7 unexg 6514 . . . 4
8 ssexg 4555 . . . 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  e.wcel 1758   cvv 3081  \cdif 3439  u.cun 3440  C_wss 3442
This theorem is referenced by:  elpwun  6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4530  ax-nul 4538  ax-pr 4648  ax-un 6505
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3752  df-sn 3994  df-pr 3996  df-uni 4209
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