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Theorem elpwuni 4053
 Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
elpwuni (B A → (A BA = B))

Proof of Theorem elpwuni
StepHypRef Expression
1 sspwuni 4051 . 2 (A BA B)
2 unissel 3920 . . . 4 ((A B B A) → A = B)
32expcom 424 . . 3 (B A → (A BA = B))
4 eqimss 3323 . . 3 (A = BA B)
53, 4impbid1 194 . 2 (B A → (A BA = B))
61, 5syl5bb 248 1 (B A → (A BA = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  ∪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 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-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-uni 3892 This theorem is referenced by: (None)
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