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Theorem unidif0 4798
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 4422 . . . 4 ((𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
2 undif1 4015 . . . . . 6 ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3 uncom 3735 . . . . . 6 (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
42, 3eqtr2i 2644 . . . . 5 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
54unieqi 4411 . . . 4 ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
6 0ex 4750 . . . . . . 7 ∅ ∈ V
76unisn 4417 . . . . . 6 {∅} = ∅
87uneq2i 3742 . . . . 5 ( (𝐴 ∖ {∅}) ∪ {∅}) = ( (𝐴 ∖ {∅}) ∪ ∅)
9 un0 3939 . . . . 5 ( (𝐴 ∖ {∅}) ∪ ∅) = (𝐴 ∖ {∅})
108, 9eqtr2i 2644 . . . 4 (𝐴 ∖ {∅}) = ( (𝐴 ∖ {∅}) ∪ {∅})
111, 5, 103eqtr4ri 2654 . . 3 (𝐴 ∖ {∅}) = ({∅} ∪ 𝐴)
12 uniun 4422 . . 3 ({∅} ∪ 𝐴) = ( {∅} ∪ 𝐴)
137uneq1i 3741 . . 3 ( {∅} ∪ 𝐴) = (∅ ∪ 𝐴)
1411, 12, 133eqtri 2647 . 2 (𝐴 ∖ {∅}) = (∅ ∪ 𝐴)
15 uncom 3735 . 2 (∅ ∪ 𝐴) = ( 𝐴 ∪ ∅)
16 un0 3939 . 2 ( 𝐴 ∪ ∅) = 𝐴
1714, 15, 163eqtri 2647 1 (𝐴 ∖ {∅}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cdif 3552  cun 3553  c0 3891  {csn 4148   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151  df-uni 4403
This theorem is referenced by:  infeq5i  8477  zornn0g  9271  basdif0  20668  tgdif0  20707  omsmeas  30163  stoweidlem57  39578
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