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Theorem disjdifprg2 31672
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 (𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem disjdifprg2
StepHypRef Expression
1 inex1g 5312 . . 3 (𝐴𝑉 → (𝐴𝐵) ∈ V)
2 elex 3491 . . 3 (𝐴𝑉𝐴 ∈ V)
3 disjdifprg 31671 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
41, 2, 3syl2anc 584 . 2 (𝐴𝑉Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
5 difin 4257 . . . . 5 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
65preq1i 4733 . . . 4 {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)}
76a1i 11 . . 3 (𝐴𝑉 → {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)})
87disjeq1d 5114 . 2 (𝐴𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥))
94, 8mpbid 231 1 (𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3473  cdif 3941  cin 3943  {cpr 4624  Disj wdisj 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4623  df-pr 4625  df-disj 5107
This theorem is referenced by:  measxun2  33039
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