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Theorem unipw 4117
 Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (The proof was shortened by Alan Sare, 28-Dec-2008.) (Contributed by SF, 14-Oct-1996.) (Revised by SF, 29-Dec-2008.)
Assertion
Ref Expression
unipw A = A

Proof of Theorem unipw
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3894 . . . 4 (x Ay(x y y A))
2 vex 2862 . . . . . . . 8 y V
32elpw 3728 . . . . . . 7 (y Ay A)
4 ssel 3267 . . . . . . 7 (y A → (x yx A))
53, 4sylbi 187 . . . . . 6 (y A → (x yx A))
65impcom 419 . . . . 5 ((x y y A) → x A)
76exlimiv 1634 . . . 4 (y(x y y A) → x A)
81, 7sylbi 187 . . 3 (x Ax A)
9 vex 2862 . . . . 5 x V
109snid 3760 . . . 4 x {x}
11 snelpwi 4116 . . . 4 (x A → {x} A)
12 elunii 3896 . . . 4 ((x {x} {x} A) → x A)
1310, 11, 12sylancr 644 . . 3 (x Ax A)
148, 13impbii 180 . 2 (x Ax A)
1514eqriv 2350 1 A = A
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ∪cuni 3891 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  df-uni 3892 This theorem is referenced by:  nnadjoinpw  4521
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