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Theorem uni0b 4495
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 4226 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
21ralbii 3009 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
3 dfss3 3625 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
4 neq0 3963 . . . 4 𝐴 = ∅ ↔ ∃𝑦 𝑦 𝐴)
5 rexcom4 3256 . . . . 5 (∃𝑥𝐴𝑦 𝑦𝑥 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
6 neq0 3963 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
76rexbii 3070 . . . . 5 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝑥)
8 eluni2 4472 . . . . . 6 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
98exbii 1814 . . . . 5 (∃𝑦 𝑦 𝐴 ↔ ∃𝑦𝑥𝐴 𝑦𝑥)
105, 7, 93bitr4ri 293 . . . 4 (∃𝑦 𝑦 𝐴 ↔ ∃𝑥𝐴 ¬ 𝑥 = ∅)
11 rexnal 3024 . . . 4 (∃𝑥𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
124, 10, 113bitri 286 . . 3 𝐴 = ∅ ↔ ¬ ∀𝑥𝐴 𝑥 = ∅)
1312con4bii 310 . 2 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
142, 3, 133bitr4ri 293 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  wss 3607  c0 3948  {csn 4210   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469
This theorem is referenced by:  uni0c  4496  uni0  4497  fin1a2lem11  9270  zornn0g  9365  0top  20835  filconn  21734  alexsubALTlem2  21899  ordcmp  32571  unisn0  39536
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