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Theorem snelpwi 4195
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
Assertion
Ref Expression
snelpwi (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwi
StepHypRef Expression
1 snssi 3722 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 elex 2741 . . 3 (𝐴𝐵𝐴 ∈ V)
3 snexg 4168 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
4 elpwg 3572 . . 3 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
52, 3, 43syl 17 . 2 (𝐴𝐵 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
61, 5mpbird 166 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 2141  Vcvv 2730  wss 3121  𝒫 cpw 3564  {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587
This theorem is referenced by:  unipw  4200  infpwfidom  7162  txdis  12992  txdis1cn  12993
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