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Theorem snss 3485
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 V
Assertion
Ref Expression
snss (A B ↔ {A} ⊆ B)

Proof of Theorem snss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elsn 3382 . . . 4 (x {A} ↔ x = A)
21imbi1i 227 . . 3 ((x {A} → x B) ↔ (x = Ax B))
32albii 1356 . 2 (x(x {A} → x B) ↔ x(x = Ax B))
4 dfss2 2928 . 2 ({A} ⊆ Bx(x {A} → x B))
5 snss.1 . . 3 A V
65clel2 2671 . 2 (A Bx(x = Ax B))
73, 4, 63bitr4ri 202 1 (A B ↔ {A} ⊆ B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  Vcvv 2551  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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373
This theorem is referenced by:  snssg  3491  prss  3511  tpss  3520  snelpw  3940  sspwb  3943  mss  3953  exss  3954  elnn  4271  relsn  4386  fnressn  5292  un0mulcl  7992  nn0ssz  8039  bdsnss  9328
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