Theorem iuniin 3980

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	⊢ ∪x ∈ A ∩y ∈ B C ⊆ ∩y ∈ B ∪x ∈ A C

Distinct variable groups:   x,y   y,A   x,B

Allowed substitution hints:   A(x)   B(y)   C(x,y)

Proof of Theorem iuniin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 2728	. . . 4 ⊢ (∃x ∈ A ∀y ∈ B z ∈ C → ∀y ∈ B ∃x ∈ A z ∈ C)
2		vex 2863	. . . . . 6 ⊢ z ∈ V
3		eliin 3975	. . . . . 6 ⊢ (z ∈ V → (z ∈ ∩y ∈ B C ↔ ∀y ∈ B z ∈ C))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (z ∈ ∩y ∈ B C ↔ ∀y ∈ B z ∈ C)
5	4	rexbii 2640	. . . 4 ⊢ (∃x ∈ A z ∈ ∩y ∈ B C ↔ ∃x ∈ A ∀y ∈ B z ∈ C)
6		eliun 3974	. . . . 5 ⊢ (z ∈ ∪x ∈ A C ↔ ∃x ∈ A z ∈ C)
7	6	ralbii 2639	. . . 4 ⊢ (∀y ∈ B z ∈ ∪x ∈ A C ↔ ∀y ∈ B ∃x ∈ A z ∈ C)
8	1, 5, 7	3imtr4i 257	. . 3 ⊢ (∃x ∈ A z ∈ ∩y ∈ B C → ∀y ∈ B z ∈ ∪x ∈ A C)
9		eliun 3974	. . 3 ⊢ (z ∈ ∪x ∈ A ∩y ∈ B C ↔ ∃x ∈ A z ∈ ∩y ∈ B C)
10		eliin 3975	. . . 4 ⊢ (z ∈ V → (z ∈ ∩y ∈ B ∪x ∈ A C ↔ ∀y ∈ B z ∈ ∪x ∈ A C))
11	2, 10	ax-mp 5	. . 3 ⊢ (z ∈ ∩y ∈ B ∪x ∈ A C ↔ ∀y ∈ B z ∈ ∪x ∈ A C)
12	8, 9, 11	3imtr4i 257	. 2 ⊢ (z ∈ ∪x ∈ A ∩y ∈ B C → z ∈ ∩y ∈ B ∪x ∈ A C)
13	12	ssriv 3278	1 ⊢ ∪x ∈ A ∩y ∈ B C ⊆ ∩y ∈ B ∪x ∈ A C

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  ∪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-ss 3260  df-iun 3972  df-iin 3973

This theorem is referenced by: (None)