Theorem disjdifprg2 32557

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 5289	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
2		elex 3480	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		disjdifprg 32556	. . 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 4247	. . . . 5 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
6	5	preq1i 4712	. . . 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 5094	. 2 ⊢ (𝐴 ∈ 𝑉 → (Disj 𝑥 ∈ {(𝐴 ∖ (𝐴 ∩ 𝐵)), (𝐴 ∩ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥))
9	4, 8	mpbid 232	1 ⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925  {cpr 4603  Disj wdisj 5086

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 2707  ax-sep 5266  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-sn 4602  df-pr 4604  df-disj 5087

This theorem is referenced by:  measxun2  34241