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Mirrors > Home > ILE Home > Th. List > iotaint | GIF version |
Description: Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
iotaint | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotauni 5165 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | uniintabim 3861 | . 2 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | eqtrd 2198 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1343 ∃!weu 2014 {cab 2151 ∪ cuni 3789 ∩ cint 3824 ℩cio 5151 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-sbc 2952 df-un 3120 df-in 3122 df-sn 3582 df-pr 3583 df-uni 3790 df-int 3825 df-iota 5153 |
This theorem is referenced by: bdcriota 13765 |
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