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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 4456 | . . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) | |
2 | undif1 4043 | . . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅}) | |
3 | uncom 3757 | . . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴) | |
4 | 2, 3 | eqtr2i 2645 | . . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅}) |
5 | 4 | unieqi 4445 | . . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅}) |
6 | 0ex 4790 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | 6 | unisn 4451 | . . . . . 6 ⊢ ∪ {∅} = ∅ |
8 | 7 | uneq2i 3764 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅) |
9 | un0 3967 | . . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅}) | |
10 | 8, 9 | eqtr2i 2645 | . . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) |
11 | 1, 5, 10 | 3eqtr4ri 2655 | . . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴) |
12 | uniun 4456 | . . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴) | |
13 | 7 | uneq1i 3763 | . . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴) |
14 | 11, 12, 13 | 3eqtri 2648 | . 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴) |
15 | uncom 3757 | . 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅) | |
16 | un0 3967 | . 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴 | |
17 | 14, 15, 16 | 3eqtri 2648 | 1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∖ cdif 3571 ∪ cun 3572 ∅c0 3915 {csn 4177 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 |
This theorem is referenced by: infeq5i 8533 zornn0g 9327 basdif0 20757 tgdif0 20796 omsmeas 30385 stoweidlem57 40274 |
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