Theorem iindif2 5079

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 5060 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 4499	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)))
2		eldif 3957	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	bicomi 223	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ 𝐶))
4	3	ralbii 3091	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
5		ralnex 3070	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		eliun 5000	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	5, 6	xchbinxr 334	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 621	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	1, 4, 8	3bitr3g 312	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)))
10		eliin 5001	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)))
11	10	elv 3478	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
12		eldif 3957	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
13	9, 11, 12	3bitr4g 313	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)))
14	13	eqrdv 2728	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ∖ cdif 3944  ∅c0 4321  ∪ ciun 4996  ∩ ciin 4997

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3474  df-dif 3950  df-nul 4322  df-iun 4998  df-iin 4999

This theorem is referenced by:  iinvdif  5082  iincld  22763  clsval2  22774  mretopd  22816  hauscmplem  23130  cmpfi  23132  sigapildsyslem  33457