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| Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.) | 
| Ref | Expression | 
|---|---|
| unisng | ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfsn2 4639 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 2 | 1 | unieqi 4919 | . . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴} | 
| 3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴}) | 
| 4 | uniprg 4923 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)) | |
| 5 | 4 | anidms 566 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)) | 
| 6 | unidm 4157 | . . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
| 7 | 6 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴) | 
| 8 | 3, 5, 7 | 3eqtrd 2781 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 ∪ 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 | 
| This theorem is referenced by: unisn 4926 unisn3 4928 dfnfc2 4929 unisn2 5312 unisucs 6461 en2other2 10049 pmtrprfv 19471 dprdsn 20056 indistopon 23008 ordtuni 23198 cmpcld 23410 ptcmplem5 24064 cldsubg 24119 icccmplem2 24845 vmappw 27159 chsupsn 31432 xrge0tsmseq 33067 cycpm2tr 33139 qustrivr 33393 esumsnf 34065 prsiga 34132 rossros 34181 cvmscld 35278 unisnif 35926 topjoin 36366 fnejoin2 36370 bj-snmoore 37114 pibt2 37418 heiborlem8 37825 sucunisn 43384 onsucunitp 43386 oaun3 43395 fourierdlem80 46201 | 
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