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Theorem sspwuni 4051
 Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
sspwuni (A BA B)

Proof of Theorem sspwuni
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . 4 x V
21elpw 3728 . . 3 (x Bx B)
32ralbii 2638 . 2 (x A x Bx A x B)
4 dfss3 3263 . 2 (A Bx A x B)
5 unissb 3921 . 2 (A Bx A x B)
63, 4, 53bitr4i 268 1 (A BA B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  ℘cpw 3722  ∪cuni 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  pwssb  4052  elpwuni  4053  rintn0  4056  qsss  5986
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