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Mirrors > Home > ILE Home > Th. List > iotauni | GIF version |
Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Ref | Expression |
---|---|
iotauni | ⊢ (∃!xφ → (℩xφ) = ∪ {x ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 1900 | . 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z)) | |
2 | iotaval 4821 | . . . 4 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z) | |
3 | uniabio 4820 | . . . 4 ⊢ (∀x(φ ↔ x = z) → ∪ {x ∣ φ} = z) | |
4 | 2, 3 | eqtr4d 2072 | . . 3 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = ∪ {x ∣ φ}) |
5 | 4 | exlimiv 1486 | . 2 ⊢ (∃z∀x(φ ↔ x = z) → (℩xφ) = ∪ {x ∣ φ}) |
6 | 1, 5 | sylbi 114 | 1 ⊢ (∃!xφ → (℩xφ) = ∪ {x ∣ φ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 = wceq 1242 ∃wex 1378 ∃!weu 1897 {cab 2023 ∪ cuni 3571 ℩cio 4808 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-sn 3373 df-pr 3374 df-uni 3572 df-iota 4810 |
This theorem is referenced by: iotaint 4823 fveu 5113 riotauni 5417 |
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