Theorem iinss1 3982

Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Assertion

Ref	Expression
iinss1	|- (A C_ B -> |^|_x e. B C C_ |^|_x e. A C)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iinss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 3331	. . 3 |- (A C_ B -> (A.x e. B y e. C -> A.x e. A y e. C))
2		vex 2863	. . . 4 |- y e. _V
3		eliin 3975	. . . 4 |- (y e. _V -> (y e. |^|_x e. B C <-> A.x e. B y e. C))
4	2, 3	ax-mp 5	. . 3 |- (y e. |^|_x e. B C <-> A.x e. B y e. C)
5		eliin 3975	. . . 4 |- (y e. _V -> (y e. |^|_x e. A C <-> A.x e. A y e. C))
6	2, 5	ax-mp 5	. . 3 |- (y e. |^|_x e. A C <-> A.x e. A y e. C)
7	1, 4, 6	3imtr4g 261	. 2 |- (A C_ B -> (y e. |^|_x e. B C -> y e. |^|_x e. A C))
8	7	ssrdv 3279	1 |- (A C_ B -> |^|_x e. B C C_ |^|_x e. A C)

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2615  _Vcvv 2860   C_ wss 3258  |^|_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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is referenced by: (None)