Theorem iindif2 4036

Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4020 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 3646	. . . 4 ⊢ (A ≠ ∅ → (∀x ∈ A (y ∈ B ∧ ¬ y ∈ C) ↔ (y ∈ B ∧ ∀x ∈ A ¬ y ∈ C)))
2		eldif 3222	. . . . . 6 ⊢ (y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ C))
3	2	bicomi 193	. . . . 5 ⊢ ((y ∈ B ∧ ¬ y ∈ C) ↔ y ∈ (B ∖ C))
4	3	ralbii 2639	. . . 4 ⊢ (∀x ∈ A (y ∈ B ∧ ¬ y ∈ C) ↔ ∀x ∈ A y ∈ (B ∖ C))
5		ralnex 2625	. . . . . 6 ⊢ (∀x ∈ A ¬ y ∈ C ↔ ¬ ∃x ∈ A y ∈ C)
6		eliun 3974	. . . . . 6 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
7	5, 6	xchbinxr 302	. . . . 5 ⊢ (∀x ∈ A ¬ y ∈ C ↔ ¬ y ∈ ∪x ∈ A C)
8	7	anbi2i 675	. . . 4 ⊢ ((y ∈ B ∧ ∀x ∈ A ¬ y ∈ C) ↔ (y ∈ B ∧ ¬ y ∈ ∪x ∈ A C))
9	1, 4, 8	3bitr3g 278	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ ∪x ∈ A C)))
10		vex 2863	. . . 4 ⊢ y ∈ V
11		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A (B ∖ C) ↔ ∀x ∈ A y ∈ (B ∖ C)))
12	10, 11	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A (B ∖ C) ↔ ∀x ∈ A y ∈ (B ∖ C))
13		eldif 3222	. . 3 ⊢ (y ∈ (B ∖ ∪x ∈ A C) ↔ (y ∈ B ∧ ¬ y ∈ ∪x ∈ A C))
14	9, 12, 13	3bitr4g 279	. 2 ⊢ (A ≠ ∅ → (y ∈ ∩x ∈ A (B ∖ C) ↔ y ∈ (B ∖ ∪x ∈ A C)))
15	14	eqrdv 2351	1 ⊢ (A ≠ ∅ → ∩x ∈ A (B ∖ C) = (B ∖ ∪x ∈ A C))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∖ cdif 3207  ∅c0 3551  ∪ciun 3970  ∩ciin 3971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iun 3972  df-iin 3973

This theorem is used by: (None)