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Theorem unidif0 5247
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
StepHypRef Expression
1 uniun 4847 . . . 4 ((𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
2 undif1 4407 . . . . . 6 ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3 uncom 4115 . . . . . 6 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
42, 3eqtr2i 2848 . . . . 5 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
54unieqi 4837 . . . 4 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
6 0ex 5197 . . . . . . 7 ∅ ∈ V
76unisn 4844 . . . . . 6 {∅} = ∅
87uneq2i 4122 . . . . 5 ( (𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ ∅)
9 un0 4327 . . . . 5 ( (𝐴 ∖ {∅}) ∪ ∅) = (𝐴 ∖ {∅})
108, 9eqtr2i 2848 . . . 4 (𝐴 ∖ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
111, 5, 103eqtr4ri 2858 . . 3 (𝐴 ∖ {∅}) = ({∅} ∪ 𝐴)
12 uniun 4847 . . 3 ({∅} ∪ 𝐴) = ( {∅} ∪ 𝐴)
137uneq1i 4121 . . 3 ( {∅} ∪ 𝐴) = (∅ ∪ 𝐴)
1411, 12, 133eqtri 2851 . 2 (𝐴 ∖ {∅}) = (∅ ∪ 𝐴)
15 uncom 4115 . 2 (∅ ∪ 𝐴) = ( 𝐴 ∪ ∅)
16 un0 4327 . 2 ( 𝐴 ∪ ∅) = 𝐴
1714, 15, 163eqtri 2851 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cdif 3916  cun 3917  c0 4276  {csn 4550   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-uni 4825
This theorem is referenced by:  infeq5i  9096  zornn0g  9925  basdif0  21565  tgdif0  21604  omsmeas  31642  stoweidlem57  42630
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