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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 4476 | . . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) | |
2 | unisng 4860 | . . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
3 | 1, 2 | eqtr4d 2862 | . . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
4 | iffalse 4479 | . . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅) | |
5 | snprc 4656 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | 5 | biimpi 218 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
7 | 6 | unieqd 4855 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
8 | uni0 4869 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 7, 8 | syl6eq 2875 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅) |
10 | 4, 9 | eqtr4d 2862 | . . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}) |
11 | 3, 10 | pm2.61i 184 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴} |
12 | 11 | eqcomi 2833 | 1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 ifcif 4470 {csn 4570 ∪ cuni 4841 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-uni 4842 |
This theorem is referenced by: dfrdg4 33416 |
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