Theorem ssiun2 4010

Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun2	|- (x e. A -> B C_ U_x e. A B)

Proof of Theorem ssiun2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 2676	. . . 4 |- ((x e. A /\ y e. B) -> E.x e. A y e. B)
2	1	ex 423	. . 3 |- (x e. A -> (y e. B -> E.x e. A y e. B))
3		eliun 3974	. . 3 |- (y e. U_x e. A B <-> E.x e. A y e. B)
4	2, 3	syl6ibr 218	. 2 |- (x e. A -> (y e. B -> y e. U_x e. A B))
5	4	ssrdv 3279	1 |- (x e. A -> B C_ U_x e. A B)

Syntax hints:   -> wi 4   e. wcel 1710  E.wrex 2616   C_ wss 3258  U_ciun 3970

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

This theorem is referenced by:  ssiun2s  4011