Theorem iindif2 5100

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 5081 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iindif2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 4524	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)))
2		eldif 3986	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	bicomi 224	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ 𝐶))
4	3	ralbii 3099	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
5		ralnex 3078	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		eliun 5019	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	5, 6	xchbinxr 335	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 622	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	1, 4, 8	3bitr3g 313	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)))
10		eliin 5020	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)))
11	10	elv 3493	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
12		eldif 3986	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
13	9, 11, 12	3bitr4g 314	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)))
14	13	eqrdv 2738	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973  ∅c0 4352  ∪ ciun 5015  ∩ ciin 5016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-nul 4353  df-iun 5017  df-iin 5018

This theorem is referenced by:  iinvdif  5103  iincld  23068  clsval2  23079  mretopd  23121  hauscmplem  23435  cmpfi  23437  sigapildsyslem  34125