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Theorem unidif0 4061
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 3338 . . . . . . 7 (𝑥𝑦 → ¬ 𝑦 = ∅)
21pm4.71i 388 . . . . . 6 (𝑥𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑦 = ∅))
32anbi1i 453 . . . . 5 ((𝑥𝑦𝑦𝐴) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
4 an32 536 . . . . 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 1569 . . 3 (∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3709 . . . 4 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})))
9 eldif 3050 . . . . . . 7 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10 velsn 3514 . . . . . . . . 9 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1110notbii 642 . . . . . . . 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 1569 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
168, 15bitri 183 . . 3 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
17 eluni 3709 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
187, 16, 173bitr4i 211 . 2 (𝑥 (𝐴 ∖ {∅}) ↔ 𝑥 𝐴)
1918eqriv 2114 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1316  wex 1453  wcel 1465  cdif 3038  c0 3333  {csn 3497   cuni 3706
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334  df-sn 3503  df-uni 3707
This theorem is referenced by: (None)
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