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Theorem uni0b 3916
 Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
Assertion
Ref Expression
uni0b (A = A {})

Proof of Theorem uni0b
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsn 3748 . . 3 (x {} ↔ x = )
21ralbii 2638 . 2 (x A x {} ↔ x A x = )
3 dfss3 3263 . 2 (A {} ↔ x A x {})
4 neq0 3560 . . . 4 A = y y A)
5 rexcom4 2878 . . . . 5 (x A y y xyx A y x)
6 neq0 3560 . . . . . 6 x = y y x)
76rexbii 2639 . . . . 5 (x A ¬ x = x A y y x)
8 eluni2 3895 . . . . . 6 (y Ax A y x)
98exbii 1582 . . . . 5 (y y Ayx A y x)
105, 7, 93bitr4ri 269 . . . 4 (y y Ax A ¬ x = )
11 rexnal 2625 . . . 4 (x A ¬ x = ↔ ¬ x A x = )
124, 10, 113bitri 262 . . 3 A = ↔ ¬ x A x = )
1312con4bii 288 . 2 (A = x A x = )
142, 3, 133bitr4ri 269 1 (A = A {})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ⊆ 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:  uni0c  3917  uni0  3918
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