Theorem iundif2 4982

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4968 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

Ref	Expression
iundif2	⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iundif2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3876	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
2	1	rexbii 3170	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3		r19.42v 3263	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
4		rexnal 3160	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
5		eliin 4909	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	5	elv 3414	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	4, 6	xchbinxr 338	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 626	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
9	2, 3, 8	3bitri 300	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
10		eliun 4908	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
11		eldif 3876	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
12	9, 10, 11	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶))
13	12	eqriv 2734	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∖ cdif 3863  ∪ ciun 4904  ∩ ciin 4905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-dif 3869  df-iun 4906  df-iin 4907

This theorem is referenced by:  iuncld  21942  pnrmopn  22240  alexsublem  22941  bcth3  24228  iundifdifd  30620  iundifdif  30621