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Theorem pwuni 4048
Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
pwuni 𝐴 ⊆ 𝒫 𝐴

Proof of Theorem pwuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elssuni 3703 . . 3 (𝑥𝐴𝑥 𝐴)
2 vex 2636 . . . 4 𝑥 ∈ V
32elpw 3455 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
41, 3sylibr 133 . 2 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
54ssriv 3043 1 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1445  wss 3013  𝒫 cpw 3449   cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-uni 3676
This theorem is referenced by:  uniexb  4323  2pwuninelg  6086  istopon  11864  eltg3i  11908  mopnfss  12233
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