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Theorem unidif0 5072
 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 4692 . . . 4 ((𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
2 undif1 4267 . . . . . 6 ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3 uncom 3980 . . . . . 6 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
42, 3eqtr2i 2803 . . . . 5 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
54unieqi 4680 . . . 4 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
6 0ex 5026 . . . . . . 7 ∅ ∈ V
76unisn 4687 . . . . . 6 {∅} = ∅
87uneq2i 3987 . . . . 5 ( (𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ ∅)
9 un0 4193 . . . . 5 ( (𝐴 ∖ {∅}) ∪ ∅) = (𝐴 ∖ {∅})
108, 9eqtr2i 2803 . . . 4 (𝐴 ∖ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
111, 5, 103eqtr4ri 2813 . . 3 (𝐴 ∖ {∅}) = ({∅} ∪ 𝐴)
12 uniun 4692 . . 3 ({∅} ∪ 𝐴) = ( {∅} ∪ 𝐴)
137uneq1i 3986 . . 3 ( {∅} ∪ 𝐴) = (∅ ∪ 𝐴)
1411, 12, 133eqtri 2806 . 2 (𝐴 ∖ {∅}) = (∅ ∪ 𝐴)
15 uncom 3980 . 2 (∅ ∪ 𝐴) = ( 𝐴 ∪ ∅)
16 un0 4193 . 2 ( 𝐴 ∪ ∅) = 𝐴
1714, 15, 163eqtri 2806 1 (𝐴 ∖ {∅}) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∖ cdif 3789   ∪ cun 3790  ∅c0 4141  {csn 4398  ∪ cuni 4671 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-nul 5025 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-sn 4399  df-pr 4401  df-uni 4672 This theorem is referenced by:  infeq5i  8830  zornn0g  9662  basdif0  21165  tgdif0  21204  omsmeas  30983  stoweidlem57  41205
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