Theorem difxp 6166

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.) (Proof shortened by Wolf Lammen, 16-May-2025.)

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 4093	. . 3 ⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷)
2		relxp 5684	. . 3 ⊢ Rel (𝐶 × 𝐷)
3		relss 5773	. . 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 5684	. . 3 ⊢ Rel ((𝐶 ∖ 𝐴) × 𝐷)
6		relxp 5684	. . 3 ⊢ Rel (𝐶 × (𝐷 ∖ 𝐵))
7		relun 5803	. . 3 ⊢ (Rel (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (Rel ((𝐶 ∖ 𝐴) × 𝐷) ∧ Rel (𝐶 × (𝐷 ∖ 𝐵))))
8	5, 6, 7	mpbir2an 724	. 2 ⊢ Rel (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))
9		ianor 997	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵))
10	9	anbi2i 635	. . . . 5 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵)))
11		andi 1025	. . . . 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 278	. . . 4 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴) ∨ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵)))
13		opelxp 5702	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
14		opelxp 5702	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
15	14	notbii 323	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
16	13, 15	anbi12i 640	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17		opelxp 5702	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ↔ (𝑥 ∈ (𝐶 ∖ 𝐴) ∧ 𝑦 ∈ 𝐷))
18		eldif 3918	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
19	18	anbi1i 636	. . . . . 6 ⊢ ((𝑥 ∈ (𝐶 ∖ 𝐴) ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐷))
20		an32 659	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴))
21	17, 19, 20	3bitri 300	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴))
22		eldif 3918	. . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ 𝐵) ↔ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵))
23	22	anbi2i 635	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐷 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵)))
24		opelxp 5702	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ (𝐷 ∖ 𝐵)))
25		anass 474	. . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵)))
26	23, 24, 25	3bitr4i 306	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵))
27	21, 26	orbi12i 928	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑥 ∈ 𝐴) ∨ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ 𝐵)))
28	12, 16, 27	3bitr4i 306	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))))
29		eldif 3918	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
30		elun 4110	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 ∖ 𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷 ∖ 𝐵))))
31	28, 29, 30	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))))
32	4, 8, 31	eqrelriiv 5781	1 ⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  ⟨cop 4600   × cxp 5664  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by:  difxp1  6167  difxp2  6168  evlslem4  22264  txcld  23797  suppovss  33063  elrgspnlem2  33594  elrgspnsubrunlem2  33599