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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 3624 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | unieqi 3837 | . 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴} |
3 | unisn.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | unipr 3841 | . 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴) |
5 | unidm 3293 | . 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
6 | 2, 4, 5 | 3eqtri 2214 | 1 ⊢ ∪ {𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 {csn 3610 {cpr 3611 ∪ cuni 3827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3616 df-pr 3617 df-uni 3828 |
This theorem is referenced by: unisng 3844 uniintsnr 3898 unisuc 4434 op1sta 5131 op2nda 5134 elxp4 5137 uniabio 5209 iotass 5216 en1bg 6830 zrhval2 13941 |
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