Theorem undif4 1744

Description: Distribute union over difference.

Assertion

Ref	Expression
undif4	⊢ ((A ∩ C) = ∅ → (A ∪ (B ∖ C)) = ((A ∪ B) ∖ C))

Proof of Theorem undif4

Step	Hyp	Ref	Expression
1		pm2.61 109	. . . . . . . 8 ⊢ ((x ∈ A → ¬ x ∈ C) → ((¬ x ∈ A → ¬ x ∈ C) → ¬ x ∈ C))
2		ax-1 3	. . . . . . . . 9 ⊢ (¬ x ∈ C → (¬ x ∈ A → ¬ x ∈ C))
3	2	a1i 7	. . . . . . . 8 ⊢ ((x ∈ A → ¬ x ∈ C) → (¬ x ∈ C → (¬ x ∈ A → ¬ x ∈ C)))
4	1, 3	impbid 397	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ C) → ((¬ x ∈ A → ¬ x ∈ C) ↔ ¬ x ∈ C))
5		df-or 197	. . . . . . 7 ⊢ ((x ∈ A ∨ ¬ x ∈ C) ↔ (¬ x ∈ A → ¬ x ∈ C))
6	4, 5	syl5bb 410	. . . . . 6 ⊢ ((x ∈ A → ¬ x ∈ C) → ((x ∈ A ∨ ¬ x ∈ C) ↔ ¬ x ∈ C))
7	6	anbi2d 468	. . . . 5 ⊢ ((x ∈ A → ¬ x ∈ C) → (((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ C)))
8		eldif 1496	. . . . . . 7 ⊢ (x ∈ (B ∖ C) ↔ (x ∈ B ∧ ¬ x ∈ C))
9	8	orbi2i 214	. . . . . 6 ⊢ ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
10		ordi 452	. . . . . 6 ⊢ ((x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)))
11	9, 10	bitr 151	. . . . 5 ⊢ ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)))
12		elun 1601	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
13	12	anbi1i 368	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ ¬ x ∈ C) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ C))
14	7, 11, 13	3bitr4g 428	. . . 4 ⊢ ((x ∈ A → ¬ x ∈ C) → ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ C)))
15		elun 1601	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ C)) ↔ (x ∈ A ∨ x ∈ (B ∖ C)))
16		eldif 1496	. . . 4 ⊢ (x ∈ ((A ∪ B) ∖ C) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ C))
17	14, 15, 16	3bitr4g 428	. . 3 ⊢ ((x ∈ A → ¬ x ∈ C) → (x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
18	17	19.20i 691	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ C) → ∀x(x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
19		disj1 1734	. 2 ⊢ ((A ∩ C) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ C))
20		dfcleq 1098	. 2 ⊢ ((A ∪ (B ∖ C)) = ((A ∪ B) ∖ C) ↔ ∀x(x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
21	18, 19, 20	3imtr4 192	1 ⊢ ((A ∩ C) = ∅ → (A ∪ (B ∖ C)) = ((A ∪ B) ∖ C))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486  ∅c0 1707

This theorem is referenced by:  phplem2 3404

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708

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