Theorem difxp 6067

Description: Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

Assertion

Ref	Expression
difxp	⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))

Proof of Theorem difxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 4066	. . 3 ⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷)
2		relxp 5607	. . 3 ⊢ Rel (𝐶 × 𝐷)
3		relss 5692	. . 3 ⊢ (((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷) → (Rel (𝐶 × 𝐷) → Rel ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵))))
4	1, 2, 3	mp2 9	. 2 ⊢ Rel ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵))
5		relxp 5607	. . 3 ⊢ Rel ((𝐶 ∖ 𝐴) × 𝐷)
6		relxp 5607	. . 3 ⊢ Rel (𝐶 × (𝐷 ∖ 𝐵))
7		relun 5721	. . 3 ⊢ (Rel (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (Rel ((𝐶 ∖ 𝐴) × 𝐷) ∧ Rel (𝐶 × (𝐷 ∖ 𝐵))))
8	5, 6, 7	mpbir2an 708	. 2 ⊢ Rel (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))
9		ianor 979	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
10	9	anbi2i 623	. . . . 5 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵)))
11		andi 1005	. . . . 5 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵)) ↔ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴) ∨ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵)))
12	10, 11	bitri 274	. . . 4 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴) ∨ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵)))
13		opelxp 5625	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
14		opelxp 5625	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
15	14	notbii 320	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
16	13, 15	anbi12i 627	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17		opelxp 5625	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ↔ (𝑥 ∈ (𝐶 ∖ 𝐴) ∧ 𝑦 ∈ 𝐷))
18		eldif 3897	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
19	18	anbi1i 624	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐶 ∖ 𝐴) ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐷))
20		an32 643	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴))
21	19, 20	bitri 274	. . . . . 6 ⊢ ((𝑥 ∈ (𝐶 ∖ 𝐴) ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴))
22	17, 21	bitri 274	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴))
23		eldif 3897	. . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ 𝐵) ↔ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵))
24	23	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐷 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵)))
25		opelxp 5625	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐷 ∖ 𝐵)))
26		anass 469	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵)))
27	24, 25, 26	3bitr4i 303	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵))
28	22, 27	orbi12i 912	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴) ∨ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵)))
29	12, 16, 28	3bitr4i 303	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))))
30		eldif 3897	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
31		elun 4083	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))))
32	29, 30, 31	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))))
33	4, 8, 32	eqrelriiv 5700	1 ⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ⟨cop 4567   × cxp 5587  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-rel 5596

This theorem is referenced by:  difxp1  6068  difxp2  6069  evlslem4  21284  txcld  22754  suppovss  31017