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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 4531 | . . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) | |
| 2 | unisng 4925 | . . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
| 3 | 1, 2 | eqtr4d 2780 | . . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
| 4 | iffalse 4534 | . . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅) | |
| 5 | snprc 4717 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 6 | 5 | biimpi 216 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 7 | 6 | unieqd 4920 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
| 8 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 9 | 7, 8 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅) |
| 10 | 4, 9 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
| 11 | 3, 10 | pm2.61i 182 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴} |
| 12 | 11 | eqcomi 2746 | 1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ifcif 4525 {csn 4626 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: dfrdg4 35952 |
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