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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 3507 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | unieqi 3712 | . 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴} |
3 | unisn.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | unipr 3716 | . 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴) |
5 | unidm 3185 | . 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
6 | 2, 4, 5 | 3eqtri 2139 | 1 ⊢ ∪ {𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ∈ wcel 1463 Vcvv 2657 ∪ cun 3035 {csn 3493 {cpr 3494 ∪ cuni 3702 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-uni 3703 |
This theorem is referenced by: unisng 3719 uniintsnr 3773 unisuc 4295 op1sta 4978 op2nda 4981 elxp4 4984 uniabio 5056 iotass 5063 en1bg 6648 |
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