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Mirrors > Home > MPE Home > Th. List > iotauni | Structured version Visualization version 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 | eu6 2659 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | iotaval 6315 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
3 | uniabio 6314 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧) | |
4 | 2, 3 | eqtr4d 2859 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
5 | 4 | exlimiv 1931 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
6 | 1, 5 | sylbi 219 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∃wex 1780 ∃!weu 2653 {cab 2799 ∪ cuni 4824 ℩cio 6298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3488 df-sbc 3764 df-un 3929 df-in 3931 df-ss 3940 df-sn 4554 df-pr 4556 df-uni 4825 df-iota 6300 |
This theorem is referenced by: iotaint 6317 iotassuni 6320 dfiota4 6333 fveu 6647 riotauni 7106 afv2eu 43527 |
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