Theorem snssi 3852

Description: The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
snssi	|- (A e. B -> {A} C_ B)

Proof of Theorem snssi

Step	Hyp	Ref	Expression
1		snssg 3844	. 2 |- (A e. B -> (A e. B <-> {A} C_ B))
2	1	ibi 232	1 |- (A e. B -> {A} C_ B)

Syntax hints:   -> wi 4   e. wcel 1710   C_ wss 3257  {csn 3737

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-sn 3741

This theorem is referenced by:  difsnid  3854  pwpw0  3855  sssn  3864  ssunsn2  3865  pwsnALT  3882  snelpwi  4116  dfiota4  4372  nnsucelrlem4  4427  ssfin  4470  fvimacnvi  5402  fsn2  5434  map0  6025  mapsn  6026  spacssnc  6284  spacind  6287  nchoicelem13  6301