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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 4042 | . . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) | |
2 | unisng 4388 | . . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
3 | 1, 2 | eqtr4d 2647 | . . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
4 | iffalse 4045 | . . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅) | |
5 | snprc 4197 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | 5 | biimpi 205 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
7 | 6 | unieqd 4382 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
8 | uni0 4401 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 7, 8 | syl6eq 2660 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅) |
10 | 4, 9 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
11 | 3, 10 | pm2.61i 175 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴} |
12 | 11 | eqcomi 2619 | 1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 {csn 4125 ∪ cuni 4372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-uni 4373 |
This theorem is referenced by: dfrdg4 31228 |
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