Theorem snss 3494

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
snss.1	|- A e. _V

Assertion

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

Proof of Theorem snss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3392	. . . 4 |- ( x e. { A } <-> x = A )
2	1	imbi1i 227	. . 3 |- ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
3	2	albii 1359	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 2934	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2677	. 2 |- ( A e. B <-> A. x ( x = A -> x e. B ) )
7	3, 4, 6	3bitr4ri 202	1 |- ( A e. B <-> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381

This theorem is referenced by:  snssg  3500  prss  3520  tpss  3529  snelpw  3949  sspwb  3952  mss  3962  exss  3963  reg2exmidlema  4259  elnn  4328  relsn  4443  fnressn  5349  un0mulcl  8216  nn0ssz  8263  bdsnss  9993