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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 4924 | . . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) | |
2 | undif1 4467 | . . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅}) | |
3 | uncom 4145 | . . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴) | |
4 | 2, 3 | eqtr2i 2753 | . . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅}) |
5 | 4 | unieqi 4911 | . . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅}) |
6 | 0ex 5297 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | unisn 4920 | . . . . . 6 ⊢ ∪ {∅} = ∅ |
8 | 7 | uneq2i 4152 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅) |
9 | un0 4382 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅}) | |
10 | 8, 9 | eqtr2i 2753 | . . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) |
11 | 1, 5, 10 | 3eqtr4ri 2763 | . . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴) |
12 | uniun 4924 | . . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴) | |
13 | 7 | uneq1i 4151 | . . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴) |
14 | 11, 12, 13 | 3eqtri 2756 | . 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴) |
15 | uncom 4145 | . 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅) | |
16 | un0 4382 | . 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴 | |
17 | 14, 15, 16 | 3eqtri 2756 | 1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3937 ∪ cun 3938 ∅c0 4314 {csn 4620 ∪ cuni 4899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-sn 4621 df-pr 4623 df-uni 4900 |
This theorem is referenced by: infeq5i 9627 zornn0g 10496 basdif0 22778 tgdif0 22817 omsmeas 33811 stoweidlem57 45258 |
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