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Theorem disjdifprg2 29275
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 4771 . . 3 (𝐴𝑉 → (𝐴𝐵) ∈ V)
2 elex 3202 . . 3 (𝐴𝑉𝐴 ∈ V)
3 disjdifprg 29274 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
41, 2, 3syl2anc 692 . 2 (𝐴𝑉Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥)
5 difin 3845 . . . . 5 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
65preq1i 4248 . . . 4 {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)}
76a1i 11 . . 3 (𝐴𝑉 → {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)} = {(𝐴𝐵), (𝐴𝐵)})
87disjeq1d 4601 . 2 (𝐴𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴𝐵)), (𝐴𝐵)}𝑥Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥))
94, 8mpbid 222 1 (𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  cin 3559  {cpr 4157  Disj wdisj 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-pr 4158  df-disj 4594
This theorem is referenced by:  measxun2  30096
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