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Theorem snelpw 3976
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 3521 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
3 snexgOLD 3962 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
41, 3ax-mp 7 . . 3 {𝐴} ∈ V
54elpw 3392 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
62, 5bitr4i 180 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  Vcvv 2574  wss 2944  𝒫 cpw 3386  {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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
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-pw 3388  df-sn 3408
This theorem is referenced by: (None)
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