Theorem iundif2 4034

Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4021 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 3222	. . . . 5 ⊢ (y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ C))
2	1	rexbii 2640	. . . 4 ⊢ (∃x ∈ A y ∈ (B ∖ C) ↔ ∃x ∈ A (y ∈ B ∧ ¬ y ∈ C))
3		r19.42v 2766	. . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ ¬ y ∈ C) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ C))
4		rexnal 2626	. . . . . 6 ⊢ (∃x ∈ A ¬ y ∈ C ↔ ¬ ∀x ∈ A y ∈ C)
5		vex 2863	. . . . . . 7 ⊢ y ∈ V
6		eliin 3975	. . . . . . 7 ⊢ (y ∈ V → (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C))
7	5, 6	ax-mp 5	. . . . . 6 ⊢ (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C)
8	4, 7	xchbinxr 302	. . . . 5 ⊢ (∃x ∈ A ¬ y ∈ C ↔ ¬ y ∈ ∩x ∈ A C)
9	8	anbi2i 675	. . . 4 ⊢ ((y ∈ B ∧ ∃x ∈ A ¬ y ∈ C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C))
10	2, 3, 9	3bitri 262	. . 3 ⊢ (∃x ∈ A y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C))
11		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A (B ∖ C) ↔ ∃x ∈ A y ∈ (B ∖ C))
12		eldif 3222	. . 3 ⊢ (y ∈ (B ∖ ∩x ∈ A C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C))
13	10, 11, 12	3bitr4i 268	. 2 ⊢ (y ∈ ∪x ∈ A (B ∖ C) ↔ y ∈ (B ∖ ∩x ∈ A C))
14	13	eqriv 2350	1 ⊢ ∪x ∈ A (B ∖ C) = (B ∖ ∩x ∈ A C)

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∖ cdif 3207  ∪ciun 3970  ∩ciin 3971

This theorem was proved from 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 theorem 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-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-iun 3972  df-iin 3973

This theorem is referenced by: (None)