Theorem elssuni 3919

Description: An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
elssuni	|- (A e. B -> A C_ U.B)

Proof of Theorem elssuni

Step	Hyp	Ref	Expression
1		ssid 3290	. 2 |- A C_ A
2		ssuni 3913	. 2 |- ((A C_ A /\ A e. B) -> A C_ U.B)
3	1, 2	mpan 651	1 |- (A e. B -> A C_ U.B)

Syntax hints:   -> wi 4   e. wcel 1710   C_ wss 3257  U.cuni 3891

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892

This theorem is referenced by:  unissel  3920  ssunieq  3924  ncssfin  6151