MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uni0b Structured version   Visualization version   GIF version

Theorem uni0b 4621
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 velsn 4350 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
21ralbii 3127 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
3 dfss3 3750 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
4 neq0 4094 . . . 4 𝐴 = ∅ ↔ ∃𝑦 𝑦 𝐴)
5 rexcom4 3378 . . . . 5 (∃𝑥𝐴𝑦 𝑦𝑥 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
6 neq0 4094 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
76rexbii 3188 . . . . 5 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝑥)
8 eluni2 4598 . . . . . 6 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
98exbii 1943 . . . . 5 (∃𝑦 𝑦 𝐴 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
105, 7, 93bitr4ri 295 . . . 4 (∃𝑦 𝑦 𝐴 ↔ ∃𝑥𝐴 ¬ 𝑥 = ∅)
11 rexnal 3141 . . . 4 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
124, 10, 113bitri 288 . . 3 𝐴 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
1312con4bii 312 . 2 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
142, 3, 133bitr4ri 295 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  wss 3732  c0 4079  {csn 4334   cuni 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-uni 4595
This theorem is referenced by:  uni0c  4622  uni0  4623  fin1a2lem11  9485  zornn0g  9580  0top  21067  filconn  21966  alexsubALTlem2  22131  ordcmp  32885  unisn0  39873
  Copyright terms: Public domain W3C validator