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Theorem unidif0 4091
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 3368 . . . . . . 7 (𝑥𝑦 → ¬ 𝑦 = ∅)
21pm4.71i 388 . . . . . 6 (𝑥𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑦 = ∅))
32anbi1i 453 . . . . 5 ((𝑥𝑦𝑦𝐴) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
4 an32 551 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
5 anass 398 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
63, 4, 53bitr2ri 208 . . . 4 ((𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥𝑦𝑦𝐴))
76exbii 1584 . . 3 (∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3739 . . . 4 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})))
9 eldif 3080 . . . . . . 7 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10 velsn 3544 . . . . . . . . 9 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1110notbii 657 . . . . . . . 8 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
1211anbi2i 452 . . . . . . 7 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
139, 12bitri 183 . . . . . 6 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
1413anbi2i 452 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
1514exbii 1584 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
168, 15bitri 183 . . 3 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
17 eluni 3739 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
187, 16, 173bitr4i 211 . 2 (𝑥 (𝐴 ∖ {∅}) ↔ 𝑥 𝐴)
1918eqriv 2136 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1331  wex 1468  wcel 1480  cdif 3068  c0 3363  {csn 3527   cuni 3736
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-sn 3533  df-uni 3737
This theorem is referenced by: (None)
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