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Theorem disjdifprg2 32074
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 5318 . . 3 (𝐴𝑉 → (𝐴𝐵) ∈ V)
2 elex 3491 . . 3 (𝐴𝑉𝐴 ∈ V)
3 disjdifprg 32073 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
41, 2, 3syl2anc 582 . 2 (𝐴𝑉Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
5 difin 4260 . . . . 5 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
65preq1i 4739 . . . 4 {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)}
76a1i 11 . . 3 (𝐴𝑉 → {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)})
87disjeq1d 5120 . 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 1539  wcel 2104  Vcvv 3472  cdif 3944  cin 3946  {cpr 4629  Disj wdisj 5112
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-disj 5113
This theorem is referenced by:  measxun2  33506
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