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Mirrors > Home > MPE Home > Th. List > Mathboxes > unisnif | Structured version Visualization version GIF version |
Description: Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
unisnif | ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4536 | . . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) | |
2 | unisng 4929 | . . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
3 | 1, 2 | eqtr4d 2777 | . . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
4 | iffalse 4539 | . . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅) | |
5 | snprc 4721 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | 5 | biimpi 216 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
7 | 6 | unieqd 4924 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
8 | uni0 4939 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 7, 8 | eqtrdi 2790 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅) |
10 | 4, 9 | eqtr4d 2777 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
11 | 3, 10 | pm2.61i 182 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴} |
12 | 11 | eqcomi 2743 | 1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 ifcif 4530 {csn 4630 ∪ cuni 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-uni 4912 |
This theorem is referenced by: dfrdg4 35932 |
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