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Theorem snss 3521
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 3419 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21imbi1i 231 . . 3 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
32albii 1375 . 2 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
4 dfss2 2961 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 snss.1 . . 3 𝐴 ∈ V
65clel2 2699 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
73, 4, 63bitr4ri 206 1 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574  wss 2944  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-sn 3408
This theorem is referenced by:  snssg  3527  prss  3547  tpss  3556  snelpw  3976  sspwb  3979  mss  3989  exss  3990  reg2exmidlema  4286  elnn  4355  relsn  4470  fnressn  5376  un0mulcl  8272  nn0ssz  8319  bdsnss  10359
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