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Mirrors > Home > MPE Home > Th. List > unidif0 | Structured version Visualization version GIF version |
Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
Ref | Expression |
---|---|
unidif0 | ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniun 4692 | . . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) | |
2 | undif1 4267 | . . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅}) | |
3 | uncom 3980 | . . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴) | |
4 | 2, 3 | eqtr2i 2803 | . . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅}) |
5 | 4 | unieqi 4680 | . . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅}) |
6 | 0ex 5026 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | unisn 4687 | . . . . . 6 ⊢ ∪ {∅} = ∅ |
8 | 7 | uneq2i 3987 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅) |
9 | un0 4193 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅}) | |
10 | 8, 9 | eqtr2i 2803 | . . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) |
11 | 1, 5, 10 | 3eqtr4ri 2813 | . . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴) |
12 | uniun 4692 | . . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴) | |
13 | 7 | uneq1i 3986 | . . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴) |
14 | 11, 12, 13 | 3eqtri 2806 | . 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴) |
15 | uncom 3980 | . 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅) | |
16 | un0 4193 | . 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴 | |
17 | 14, 15, 16 | 3eqtri 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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