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Theorem snelpw 3949
 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 3494 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
3 snexgOLD 3935 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
41, 3ax-mp 7 . . 3 {𝐴} ∈ V
54elpw 3365 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
62, 5bitr4i 176 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359  {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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 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-pw 3361  df-sn 3381 This theorem is referenced by: (None)
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