Theorem iundif2 4999

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4986 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 3949	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
2	1	rexbii 3250	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3		r19.42v 3353	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
4		rexnal 3241	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
5		eliin 4927	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	5	elv 3502	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	4, 6	xchbinxr 337	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 624	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
9	2, 3, 8	3bitri 299	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
10		eliun 4926	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
11		eldif 3949	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
12	9, 10, 11	3bitr4i 305	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶))
13	12	eqriv 2821	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ∖ cdif 3936  ∪ ciun 4922  ∩ ciin 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-v 3499  df-dif 3942  df-iun 4924  df-iin 4925

This theorem is referenced by:  iuncld  21656  pnrmopn  21954  alexsublem  22655  bcth3  23937  iundifdifd  30316  iundifdif  30317