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Theorem disjdifprg2 23971
 Description: A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
disjdifprg2 Disj
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem disjdifprg2
StepHypRef Expression
1 inex1g 4306 . . 3
2 elex 2924 . . 3
3 disjdifprg 23970 . . 3 Disj
41, 2, 3syl2anc 643 . 2 Disj
5 difin 3538 . . . . 5
65preq1i 3846 . . . 4
76a1i 11 . . 3
87disjeq1d 4150 . 2 Disj Disj
94, 8mpbid 202 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649   wcel 1721  cvv 2916   cdif 3277   cin 3279  cpr 3775  Disj wdisj 4142 This theorem is referenced by:  measxun2  24517 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-pr 3781  df-disj 4143
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