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Mirrors > Home > ILE Home > Th. List > uniintabim | GIF version |
Description: The union and the intersection of a class abstraction are equal if there is a unique satisfying value of 𝜑(𝑥). (Contributed by Jim Kingdon, 14-Aug-2018.) |
Ref | Expression |
---|---|
uniintabim | ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3652 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | uniintsnr 3867 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∃wex 1485 ∃!weu 2019 {cab 2156 {csn 3583 ∪ cuni 3796 ∩ cint 3831 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 |
This theorem is referenced by: iotaint 5173 |
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