Theorem iindif2 2033

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 2026 to recover Enderton's theorem.

Assertion

Ref	Expression
iindif2	⊢ (¬ A = ∅ → ∩x ∈ A (B ∖ C) = (B ∖ ∪x ∈ A C))

Distinct variable group(s):   x,A   x,B

Proof of Theorem iindif2

Step	Hyp	Ref	Expression
1		r19.28zv 1769	. . . 4 ⊢ (¬ A = ∅ → (∀x ∈ A (y ∈ B ∧ ¬ y ∈ C) ↔ (y ∈ B ∧ ∀x ∈ A ¬ y ∈ C)))
2		eldif 1496	. . . . 5 ⊢ (y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ C))
3	2	biral 1223	. . . 4 ⊢ (∀x ∈ A y ∈ (B ∖ C) ↔ ∀x ∈ A (y ∈ B ∧ ¬ y ∈ C))
4		eliun 1998	. . . . . . 7 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
5	4	negbii 162	. . . . . 6 ⊢ (¬ y ∈ ∪x ∈ A C ↔ ¬ ∃x ∈ A y ∈ C)
6		ralnex 1209	. . . . . 6 ⊢ (∀x ∈ A ¬ y ∈ C ↔ ¬ ∃x ∈ A y ∈ C)
7	5, 6	bitr4 154	. . . . 5 ⊢ (¬ y ∈ ∪x ∈ A C ↔ ∀x ∈ A ¬ y ∈ C)
8	7	anbi2i 367	. . . 4 ⊢ ((y ∈ B ∧ ¬ y ∈ ∪x ∈ A C) ↔ (y ∈ B ∧ ∀x ∈ A ¬ y ∈ C))
9	1, 3, 8	3bitr4g 428	. . 3 ⊢ (¬ A = ∅ → (∀x ∈ A y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ ∪x ∈ A C)))
10		visset 1350	. . . 4 ⊢ y ∈ V
11		eliin 1999	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A (B ∖ C) ↔ ∀x ∈ A y ∈ (B ∖ C)))
12	10, 11	ax-mp 6	. . 3 ⊢ (y ∈ ∩x ∈ A (B ∖ C) ↔ ∀x ∈ A y ∈ (B ∖ C))
13		eldif 1496	. . 3 ⊢ (y ∈ (B ∖ ∪x ∈ A C) ↔ (y ∈ B ∧ ¬ y ∈ ∪x ∈ A C))
14	9, 12, 13	3bitr4g 428	. 2 ⊢ (¬ A = ∅ → (y ∈ ∩x ∈ A (B ∖ C) ↔ y ∈ (B ∖ ∪x ∈ A C)))
15	14	cleqrd 1100	1 ⊢ (¬ A = ∅ → ∩x ∈ A (B ∖ C) = (B ∖ ∪x ∈ A C))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ∖ cdif 1484  ∅c0 1707  ∪ciun 1994  ∩ciin 1995

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-rex 1206  df-v 1349  df-dif 1489  df-nul 1708  df-iun 1996  df-iin 1997

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