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Theorem snelpw 4146
 Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
Hypothesis
Ref Expression
snelpw.1 𝐴 ∈ V
Assertion
Ref Expression
snelpw (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3 𝐴 ∈ V
21snss 3659 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
31snex 4119 . . 3 {𝐴} ∈ V
43elpw 3523 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
52, 4bitr4i 186 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∈ wcel 1481  Vcvv 2691   ⊆ wss 3078  𝒫 cpw 3517  {csn 3534 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1738  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540 This theorem is referenced by: (None)
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