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Theorem snelpwg 4114
 Description: A singleton of a set belongs to a power class of a set containing it. (Contributed by SF, 1-Feb-2015.)
Assertion
Ref Expression
snelpwg (A V → ({A} BA B))

Proof of Theorem snelpwg
StepHypRef Expression
1 snssg 3844 . 2 (A V → (A B ↔ {A} B))
2 snex 4111 . . 3 {A} V
32elpw 3728 . 2 ({A} B ↔ {A} B)
41, 3syl6rbbr 255 1 (A V → ({A} BA B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741 This theorem is referenced by:  snelpw  4115
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