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Theorem snss 3561
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 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)

Proof of Theorem snss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 velsn 3458 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21imbi1i 236 . . 3 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
32albii 1404 . 2 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
4 dfss2 3012 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 snss.1 . . 3 𝐴 ∈ V
65clel2 2748 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
73, 4, 63bitr4ri 211 1 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wcel 1438  Vcvv 2619  wss 2997  {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-sn 3447
This theorem is referenced by:  snssg  3568  prss  3588  tpss  3597  snelpw  4031  sspwb  4034  mss  4044  exss  4045  reg2exmidlema  4340  elnn  4410  relsn  4531  fnressn  5467  un0mulcl  8677  nn0ssz  8738  fimaxre2  10622  fsum2dlemstep  10791  fsumabs  10822  fsumiun  10833  bdsnss  11421
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