Theorem ss2iun 4346

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun

Proof of Theorem ss2iun

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3497	. . . . 5
2	1	ralimi 2850	. . . 4
3		rexim 2922	. . . 4
4	2, 3	syl 16	. . 3
5		eliun 4335	. . 3
6		eliun 4335	. . 3
7	4, 5, 6	3imtr4g 270	. 2
8	7	ssrdv 3509	1

Syntax hints:  ->wi 4  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  U_ciun 4330

This theorem is referenced by:  iuneq2  4347  oawordri  7218  omwordri  7240  oewordri  7260  oeworde  7261  r1val1  8225  cfslb2n  8669  imasaddvallem  14926  dprdss  17076  tgcmp  19901  txcmplem1  20142  txcmplem2  20143  xkococnlem  20160  alexsubALT  20551  ptcmplem3  20554  metnrmlem2  21364  uniiccvol  21989  dvfval  22301  filnetlem3  30198  sstotbnd2  30270  equivtotbnd  30274  bnj1145  34049  bnj1136  34053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iun 4332