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| Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.) | 
| Ref | Expression | 
|---|---|
| unisn.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| unisn | ⊢ ∪ {𝐴} = 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unisn.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | unisng 4925 | . 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ∪ {𝐴} = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {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-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: unisnv 4927 unidif0 5360 op1sta 6245 op2nda 6248 opswap 6249 fvssunirnOLD 6940 funfv 6996 dffv2 7004 nlim1 8527 tc2 9782 cflim2 10303 fin1a2lem12 10451 acsmapd 18599 ghmqusnsglem1 19298 ghmquskerlem1 19301 pmtrprfval 19505 lspuni0 21008 lss0v 21015 zrhval2 21519 indistopon 23008 refun0 23523 qtopeu 23724 hmphindis 23805 filconn 23891 ufildr 23939 cnextfres1 24076 bday1s 27876 old1 27914 madeoldsuc 27923 zs12bday 28424 dimval 33651 dimvalfi 33652 locfinref 33840 pstmfval 33895 esumval 34047 esumpfinval 34076 esumpfinvalf 34077 prsiga 34132 carsggect 34320 indispconn 35239 onsucsuccmpi 36444 bj-nuliotaALT 37059 heiborlem3 37820 isomenndlem 46545 uniimaelsetpreimafv 47383 | 
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