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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 2029 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | iotaval 5184 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
3 | uniabio 5183 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧) | |
4 | 2, 3 | eqtr4d 2213 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
5 | 4 | exlimiv 1598 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
6 | 1, 5 | sylbi 121 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∃!weu 2026 {cab 2163 ∪ cuni 3807 ℩cio 5171 |
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-eu 2029 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-sn 3597 df-pr 3598 df-uni 3808 df-iota 5173 |
This theorem is referenced by: iotaint 5186 fveu 5502 riotauni 5830 |
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