Theorem snssi 3853

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 3845	. 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 3258  {csn 3738

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-sn 3742

This theorem is referenced by:  difsnid  3855  pwpw0  3856  sssn  3865  ssunsn2  3866  pwsnALT  3883  snelpwi  4117  dfiota4  4373  nnsucelrlem4  4428  ssfin  4471  fvimacnvi  5403  fsn2  5435  map0  6026  mapsn  6027  spacssnc  6285  spacind  6288  nchoicelem13  6302