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Theorem uni0b 4772
 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 4490 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
21ralbii 3131 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
3 dfss3 3880 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
4 neq0 4231 . . . 4 𝐴 = ∅ ↔ ∃𝑦 𝑦 𝐴)
5 rexcom4 3212 . . . . 5 (∃𝑥𝐴𝑦 𝑦𝑥 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
6 neq0 4231 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
76rexbii 3210 . . . . 5 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝑥)
8 eluni2 4751 . . . . . 6 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
98exbii 1830 . . . . 5 (∃𝑦 𝑦 𝐴 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
105, 7, 93bitr4ri 305 . . . 4 (∃𝑦 𝑦 𝐴 ↔ ∃𝑥𝐴 ¬ 𝑥 = ∅)
11 rexnal 3201 . . . 4 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
124, 10, 113bitri 298 . . 3 𝐴 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
1312con4bii 322 . 2 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
142, 3, 133bitr4ri 305 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1522  ∃wex 1762   ∈ wcel 2080  ∀wral 3104  ∃wrex 3105   ⊆ wss 3861  ∅c0 4213  {csn 4474  ∪ cuni 4747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-v 3438  df-dif 3864  df-in 3868  df-ss 3876  df-nul 4214  df-sn 4475  df-uni 4748 This theorem is referenced by:  uni0c  4773  uni0  4774  fin1a2lem11  9681  zornn0g  9776  0top  21275  filconn  22175  alexsubALTlem2  22340  ordcmp  33398  unisn0  40868
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