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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-iotaex | Structured version Visualization version GIF version |
Description: iotaex 6408 without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
sn-iotaex | ⊢ (℩𝑥𝜑) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotavallem 40179 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) | |
2 | vex 3435 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | eqeltrdi 2847 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V) |
4 | 3 | exlimiv 1933 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V) |
5 | sn-iotanul 40181 | . . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅) | |
6 | 0ex 5231 | . . 3 ⊢ ∅ ∈ V | |
7 | 5, 6 | eqeltrdi 2847 | . 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V) |
8 | 4, 7 | pm2.61i 182 | 1 ⊢ (℩𝑥𝜑) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 Vcvv 3431 ∅c0 4258 {csn 4563 ℩cio 6384 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-sn 4564 df-pr 4566 df-uni 4842 df-iota 6386 |
This theorem is referenced by: (None) |
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