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Theorem uni0b 3652
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 eq0 3284 . . . 4 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
21ralbii 2378 . . 3 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑥𝐴𝑦 ¬ 𝑦𝑥)
3 ralcom4 2632 . . 3 (∀𝑥𝐴𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
42, 3bitri 182 . 2 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
5 dfss3 3000 . . 3 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
6 velsn 3439 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
76ralbii 2378 . . 3 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
85, 7bitri 182 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
9 eluni2 3631 . . . . 5 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
109notbii 627 . . . 4 𝑦 𝐴 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1110albii 1400 . . 3 (∀𝑦 ¬ 𝑦 𝐴 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
12 eq0 3284 . . 3 ( 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 𝐴)
13 ralnex 2363 . . . 4 (∀𝑥𝐴 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1413albii 1400 . . 3 (∀𝑦𝑥𝐴 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
1511, 12, 143bitr4i 210 . 2 ( 𝐴 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
164, 8, 153bitr4ri 211 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wal 1283   = wceq 1285  wcel 1434  wral 2353  wrex 2354  wss 2984  c0 3269  {csn 3422   cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-uni 3628
This theorem is referenced by:  uni0c  3653  uni0  3654
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