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Theorem unidif0 3947
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 3256 . . . . . . 7 (𝑥𝑦 → ¬ 𝑦 = ∅)
21pm4.71i 377 . . . . . 6 (𝑥𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑦 = ∅))
32anbi1i 439 . . . . 5 ((𝑥𝑦𝑦𝐴) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
4 an32 504 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦𝐴))
5 anass 387 . . . . 5 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
63, 4, 53bitr2ri 202 . . . 4 ((𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥𝑦𝑦𝐴))
76exbii 1512 . . 3 (∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3610 . . . 4 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})))
9 eldif 2954 . . . . . . 7 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10 velsn 3419 . . . . . . . . 9 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1110notbii 604 . . . . . . . 8 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
1211anbi2i 438 . . . . . . 7 ((𝑦𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
139, 12bitri 177 . . . . . 6 (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦𝐴 ∧ ¬ 𝑦 = ∅))
1413anbi2i 438 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
1514exbii 1512 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
168, 15bitri 177 . . 3 (𝑥 (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦 = ∅)))
17 eluni 3610 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
187, 16, 173bitr4i 205 . 2 (𝑥 (𝐴 ∖ {∅}) ↔ 𝑥 𝐴)
1918eqriv 2053 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101   = wceq 1259  wex 1397  wcel 1409  cdif 2941  c0 3251  {csn 3402   cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-nul 3252  df-sn 3408  df-uni 3608
This theorem is referenced by: (None)
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