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Theorem uni0b 3756
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 3376 . . . 4 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
21ralbii 2439 . . 3 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑥𝐴𝑦 ¬ 𝑦𝑥)
3 ralcom4 2703 . . 3 (∀𝑥𝐴𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
42, 3bitri 183 . 2 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
5 dfss3 3082 . . 3 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
6 velsn 3539 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
76ralbii 2439 . . 3 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
85, 7bitri 183 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
9 eluni2 3735 . . . . 5 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
109notbii 657 . . . 4 𝑦 𝐴 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1110albii 1446 . . 3 (∀𝑦 ¬ 𝑦 𝐴 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
12 eq0 3376 . . 3 ( 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 𝐴)
13 ralnex 2424 . . . 4 (∀𝑥𝐴 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1413albii 1446 . . 3 (∀𝑦𝑥𝐴 ¬ 𝑦𝑥 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
1511, 12, 143bitr4i 211 . 2 ( 𝐴 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
164, 8, 153bitr4ri 212 1 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1329   = wceq 1331  wcel 1480  wral 2414  wrex 2415  wss 3066  c0 3358  {csn 3522   cuni 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-uni 3732
This theorem is referenced by:  uni0c  3757  uni0  3758
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