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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iotan0aiotaex | Structured version Visualization version GIF version | ||
| Description: If the iota over a wff 𝜑 is not empty, the alternate iota over 𝜑 is a set. (Contributed by AV, 25-Aug-2022.) |
| Ref | Expression |
|---|---|
| iotan0aiotaex | ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iotanul 6539 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
| 2 | 1 | necon1ai 2968 | . 2 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑) |
| 3 | aiotaexb 47101 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
| 4 | 2, 3 | sylib 218 | 1 ⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∃!weu 2568 ≠ wne 2940 Vcvv 3480 ∅c0 4333 ℩cio 6512 ℩'caiota 47095 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 df-int 4947 df-iota 6514 df-aiota 47097 |
| This theorem is referenced by: (None) |
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