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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 6514 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
2 | 1 | eqcomd 2738 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
3 | 2 | eximi 1837 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑)) |
4 | eu6 2568 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
5 | isset 3487 | . 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 291 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃!weu 2562 Vcvv 3474 ℩cio 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 |
This theorem is referenced by: iotasbc 43168 pm14.18 43177 iotavalb 43179 sbiota1 43183 |
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