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	⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem disjdifprg2

Step	Hyp	Ref	Expression
1		inex1g 5312	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
2		elex 3491	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		disjdifprg 31671	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐴 ∈ V) → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥)
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥)
5		difin 4257	. . . . 5 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
6	5	preq1i 4733	. . . 4 ⊢ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}
7	6	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
8	7	disjeq1d 5114	. 2 ⊢ (𝐴 ∈ 𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥))
9	4, 8	mpbid 231	1 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)

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