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Theorem snssg 3527
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, 22-Jul-2001.)
Assertion
Ref Expression
snssg (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))

Proof of Theorem snssg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2116 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
2 sneq 3413 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
32sseq1d 2999 . 2 (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4 vex 2577 . . 3 𝑥 ∈ V
54snss 3521 . 2 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
61, 3, 5vtoclbg 2631 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  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:  snssi  3535  snssd  3536  prssg  3548  ordtri2orexmid  4275  ordtri2or2exmid  4323  fvimacnvi  5308  fvimacnv  5309
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