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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 3603 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | unieqd 3820 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴}) |
3 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2192 | . 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴)) |
5 | vex 2740 | . . 3 ⊢ 𝑥 ∈ V | |
6 | 5 | unisn 3825 | . 2 ⊢ ∪ {𝑥} = 𝑥 |
7 | 4, 6 | vtoclg 2797 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 {csn 3592 ∪ cuni 3809 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-uni 3810 |
This theorem is referenced by: dfnfc2 3827 unisucg 4413 unisn3 4444 opswapg 5113 funfvdm 5577 en2other2 7191 |
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