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Theorem uni0c 3917
 Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
uni0c (A = x A x = )
Distinct variable group:   x,A

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 3916 . 2 (A = A {})
2 dfss3 3263 . 2 (A {} ↔ x A x {})
3 elsn 3748 . . 3 (x {} ↔ x = )
43ralbii 2638 . 2 (x A x {} ↔ x A x = )
51, 2, 43bitri 262 1 (A = x A x = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  ∅c0 3550  {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 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892 This theorem is referenced by: (None)
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