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Mirrors > Home > ILE Home > Th. List > uniintsnr | GIF version |
Description: The union and intersection of a singleton are equal. See also eusn 3605. (Contributed by Jim Kingdon, 14-Aug-2018.) |
Ref | Expression |
---|---|
uniintsnr | ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | unisn 3760 | . . 3 ⊢ ∪ {𝑥} = 𝑥 |
3 | unieq 3753 | . . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
4 | inteq 3782 | . . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥}) | |
5 | 1 | intsn 3814 | . . . 4 ⊢ ∩ {𝑥} = 𝑥 |
6 | 4, 5 | eqtrdi 2189 | . . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥) |
7 | 2, 3, 6 | 3eqtr4a 2199 | . 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴) |
8 | 7 | exlimiv 1578 | 1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∃wex 1469 {csn 3532 ∪ cuni 3744 ∩ cint 3779 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 |
This theorem is referenced by: uniintabim 3816 |
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