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Theorem unidif0 4987
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 4608 . . . 4 ((𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
2 undif1 4187 . . . . . 6 ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3 uncom 3900 . . . . . 6 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
42, 3eqtr2i 2783 . . . . 5 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
54unieqi 4597 . . . 4 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
6 0ex 4942 . . . . . . 7 ∅ ∈ V
76unisn 4603 . . . . . 6 {∅} = ∅
87uneq2i 3907 . . . . 5 ( (𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ ∅)
9 un0 4110 . . . . 5 ( (𝐴 ∖ {∅}) ∪ ∅) = (𝐴 ∖ {∅})
108, 9eqtr2i 2783 . . . 4 (𝐴 ∖ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
111, 5, 103eqtr4ri 2793 . . 3 (𝐴 ∖ {∅}) = ({∅} ∪ 𝐴)
12 uniun 4608 . . 3 ({∅} ∪ 𝐴) = ( {∅} ∪ 𝐴)
137uneq1i 3906 . . 3 ( {∅} ∪ 𝐴) = (∅ ∪ 𝐴)
1411, 12, 133eqtri 2786 . 2 (𝐴 ∖ {∅}) = (∅ ∪ 𝐴)
15 uncom 3900 . 2 (∅ ∪ 𝐴) = ( 𝐴 ∪ ∅)
16 un0 4110 . 2 ( 𝐴 ∪ ∅) = 𝐴
1714, 15, 163eqtri 2786 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cdif 3712  cun 3713  c0 4058  {csn 4321   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by:  infeq5i  8706  zornn0g  9519  basdif0  20959  tgdif0  20998  omsmeas  30694  stoweidlem57  40777
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