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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iotaexeu | Structured version Visualization version GIF version | ||
| Description: The iota class exists. This theorem does not require ax-nul 5306 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
| Ref | Expression |
|---|---|
| iotaexeu | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iotaval 6532 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
| 2 | 1 | eqcomd 2743 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
| 3 | 2 | eximi 1835 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑)) |
| 4 | eu6 2574 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 5 | isset 3494 | . 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑)) | |
| 6 | 3, 4, 5 | 3imtr4i 292 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 ℩cio 6512 |
| 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-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 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-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: iotasbc 44438 pm14.18 44447 iotavalb 44449 sbiota1 44453 |
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