Theorem iunss2 4375

Description: A subclass condition on the members of two indexed classes (x) and () that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4282. (Contributed by NM, 9-Dec-2004.)

Assertion

Ref	Expression
iunss2

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem iunss2

Step	Hyp	Ref	Expression
1		ssiun 4372	. . 3
2	1	ralimi 2850	. 2
3		iunss 4371	. 2
4	2, 3	sylibr 212	1

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

This theorem is referenced by:  iunxdif2  4378  oaass  7229  odi  7247  omass  7248  oelim2  7263

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-or 370  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