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Theorem disjdifprg2 32589
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 5319 . . 3 (𝐴𝑉 → (𝐴𝐵) ∈ V)
2 elex 3501 . . 3 (𝐴𝑉𝐴 ∈ V)
3 disjdifprg 32588 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
41, 2, 3syl2anc 584 . 2 (𝐴𝑉Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
5 difin 4272 . . . . 5 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
65preq1i 4736 . . . 4 {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)}
76a1i 11 . . 3 (𝐴𝑉 → {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)})
87disjeq1d 5118 . 2 (𝐴𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥))
94, 8mpbid 232 1 (𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  cin 3950  {cpr 4628  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-disj 5111
This theorem is referenced by:  measxun2  34211
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