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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 4535 | . . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) | |
2 | unisng 4930 | . . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
3 | 1, 2 | eqtr4d 2776 | . . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
4 | iffalse 4538 | . . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅) | |
5 | snprc 4722 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | 5 | biimpi 215 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
7 | 6 | unieqd 4923 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
8 | uni0 4940 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 7, 8 | eqtrdi 2789 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅) |
10 | 4, 9 | eqtr4d 2776 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
11 | 3, 10 | pm2.61i 182 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴} |
12 | 11 | eqcomi 2742 | 1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 ifcif 4529 {csn 4629 ∪ cuni 4909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-uni 4910 |
This theorem is referenced by: dfrdg4 34923 |
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