Theorem intssuni 4309

Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Assertion

Ref	Expression
intssuni

Proof of Theorem intssuni

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.2z 3918	. . . 4
2	1	ex 434	. . 3
3		vex 3112	. . . 4
4	3	elint2 4293	. . 3
5		eluni2 4253	. . 3
6	2, 4, 5	3imtr4g 270	. 2
7	6	ssrdv 3509	1

Syntax hints:  ->wi 4  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  C_wss 3475  c0 3784  U.cuni 4249  |^|cint 4286

This theorem is referenced by:  unissint  4311  intssuni2  4312  fin23lem31  8744  wunint  9114  tskint  9184  incexc  13649  incexc2  13650  subgint  16225  efgval  16735  lbsextlem3  17806  cssmre  18724  uffixfr  20424  uffix2  20425  uffixsn  20426  insiga  28137  dfon2lem8  29222  intidl  30426  elrfi  30626

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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-uni 4250  df-int 4287