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Mirrors > Home > ILE Home > Th. List > unisn | GIF version |
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 | dfsn2 3584 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | unieqi 3793 | . 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴} |
3 | unisn.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | unipr 3797 | . 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴) |
5 | unidm 3260 | . 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
6 | 2, 4, 5 | 3eqtri 2189 | 1 ⊢ ∪ {𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 {csn 3570 {cpr 3571 ∪ cuni 3783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-uni 3784 |
This theorem is referenced by: unisng 3800 uniintsnr 3854 unisuc 4385 op1sta 5079 op2nda 5082 elxp4 5085 uniabio 5157 iotass 5164 en1bg 6757 |
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