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Mirrors > Home > ILE Home > Th. List > unisng | GIF version |
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 | sneq 3386 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | unieqd 3591 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴}) |
3 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2054 | . 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴)) |
5 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
6 | 5 | unisn 3596 | . 2 ⊢ ∪ {𝑥} = 𝑥 |
7 | 4, 6 | vtoclg 2613 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 {csn 3375 ∪ cuni 3580 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-uni 3581 |
This theorem is referenced by: dfnfc2 3598 unisucg 4151 unisn3 4180 opswapg 4807 funfvdm 5236 |
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