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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 ∈ 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 3392 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21imbi1i 227 . . 3 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
32albii 1359 . 2 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
4 dfss2 2934 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 snss.1 . . 3 𝐴 ∈ V
65clel2 2677 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
73, 4, 63bitr4ri 202 1 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ 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
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