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Mirrors > Home > MPE Home > Th. List > iotaexOLD | Structured version Visualization version GIF version |
Description: Obsolete version of iotaex 6513 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iotaexOLD | ⊢ (℩𝑥𝜑) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 6511 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
2 | 1 | eqcomd 2739 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
3 | 2 | eximi 1838 | . . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑)) |
4 | eu6 2569 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
5 | isset 3488 | . . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 292 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
7 | iotanul 6518 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
8 | 0ex 5306 | . . 3 ⊢ ∅ ∈ V | |
9 | 7, 8 | eqeltrdi 2842 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
10 | 6, 9 | pm2.61i 182 | 1 ⊢ (℩𝑥𝜑) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃!weu 2563 Vcvv 3475 ∅c0 4321 ℩cio 6490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6492 |
This theorem is referenced by: (None) |
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