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Theorem disjdifprg2 30342
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 5190 . . 3 (𝐴𝑉 → (𝐴𝐵) ∈ V)
2 elex 3462 . . 3 (𝐴𝑉𝐴 ∈ V)
3 disjdifprg 30341 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
41, 2, 3syl2anc 587 . 2 (𝐴𝑉Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
5 difin 4191 . . . . 5 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
65preq1i 4635 . . . 4 {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)}
76a1i 11 . . 3 (𝐴𝑉 → {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)})
87disjeq1d 5006 . 2 (𝐴𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥))
94, 8mpbid 235 1 (𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444  cdif 3881  cin 3883  {cpr 4530  Disj wdisj 4998
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-disj 4999
This theorem is referenced by:  measxun2  31577
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