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Theorem unidif0 4002
Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif0 (𝐴 ∖ {∅}) = 𝐴

Proof of Theorem unidif0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3291 . . . . . . 7 (𝑥𝑦 → ¬ 𝑦 = ∅)
21pm4.71i 383 . . . . . 6 (𝑥𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑦 = ∅))
32anbi1i 446 . . . . 5 ((𝑥𝑦𝑦𝐴) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
4 an32 529 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
5 anass 393 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
63, 4, 53bitr2ri 207 . . . 4 ((𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥𝑦𝑦𝐴))
76exbii 1541 . . 3 (∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3656 . . . 4 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})))
9 eldif 3008 . . . . . . 7 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10 velsn 3463 . . . . . . . . 9 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1110notbii 629 . . . . . . . 8 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
1211anbi2i 445 . . . . . . 7 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
139, 12bitri 182 . . . . . 6 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
1413anbi2i 445 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
1514exbii 1541 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
168, 15bitri 182 . . 3 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
17 eluni 3656 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
187, 16, 173bitr4i 210 . 2 (𝑥 (𝐴 ∖ {∅}) ↔ 𝑥 𝐴)
1918eqriv 2085 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1289  wex 1426  wcel 1438  cdif 2996  c0 3286  {csn 3446   cuni 3653
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3452  df-uni 3654
This theorem is referenced by: (None)
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