ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uni0b GIF version

Theorem uni0b 3700
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 3320 . . . 4 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
21ralbii 2395 . . 3 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑥𝐴𝑦 ¬ 𝑦𝑥)
3 ralcom4 2655 . . 3 (∀𝑥𝐴𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
42, 3bitri 183 . 2 (∀𝑥𝐴 𝑥 = ∅ ↔ ∀𝑦𝑥𝐴 ¬ 𝑦𝑥)
5 dfss3 3029 . . 3 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
6 velsn 3483 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
76ralbii 2395 . . 3 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
85, 7bitri 183 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
9 eluni2 3679 . . . . 5 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
109notbii 632 . . . 4 𝑦 𝐴 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1110albii 1411 . . 3 (∀𝑦 ¬ 𝑦 𝐴 ↔ ∀𝑦 ¬ ∃𝑥𝐴 𝑦𝑥)
12 eq0 3320 . . 3 ( 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 𝐴)
13 ralnex 2380 . . . 4 (∀𝑥𝐴 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝐴 𝑦𝑥)
1413albii 1411 . . 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 1294   = wceq 1296  wcel 1445  wral 2370  wrex 2371  wss 3013  c0 3302  {csn 3466   cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-uni 3676
This theorem is referenced by:  uni0c  3701  uni0  3702
  Copyright terms: Public domain W3C validator