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Theorem uni0b 3821
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 3433 . . . 4 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
21ralbii 2476 . . 3 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑥𝐴𝑦 ¬ 𝑦𝑥)
3 ralcom4 2752 . . 3 (∀𝑥𝐴𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
42, 3bitri 183 . 2 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
5 dfss3 3137 . . 3 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
6 velsn 3600 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
76ralbii 2476 . . 3 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
85, 7bitri 183 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
9 eluni2 3800 . . . . 5 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
109notbii 663 . . . 4 𝑦 𝐴 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1110albii 1463 . . 3 (∀𝑦 ¬ 𝑦 𝐴 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
12 eq0 3433 . . 3 ( 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 𝐴)
13 ralnex 2458 . . . 4 (∀𝑥𝐴 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1413albii 1463 . . 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 1346   = wceq 1348  wcel 2141  wral 2448  wrex 2449  wss 3121  c0 3414  {csn 3583   cuni 3796
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797
This theorem is referenced by:  uni0c  3822  uni0  3823
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