Theorem snssd 3500

Description: The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
snssd.1	|- (ph -> A e. B)

Assertion

Ref	Expression
snssd	|- (ph -> {A } C_ B)

Proof of Theorem snssd

Step	Hyp	Ref	Expression
1		snssd.1	. 2 |- (ph -> A e. B)
2		snssg 3491	. . 3 |- (A e. B -> (A e. B <-> {A } C_ B))
3	1, 2	syl 14	. 2 |- (ph -> (A e. B <-> {A } C_ B))
4	1, 3	mpbid 135	1 |- (ph -> {A } C_ B)

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1390   C_ wss 2911   {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373

This theorem is referenced by:  ecinxp  6117  xpdom3m  6244  un0addcl  7991  un0mulcl  7992  fseq1p1m1  8726  bj-omtrans  9416