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Mirrors > Home > MPE Home > Th. List > iotaex | Structured version Visualization version GIF version |
Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
iotaex | ⊢ (℩𝑥𝜑) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval2 6461 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) | |
2 | vex 3447 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | eqeltrdi 2846 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V) |
4 | 3 | exlimiv 1933 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V) |
5 | iotanul2 6463 | . . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅) | |
6 | 0ex 5262 | . . 3 ⊢ ∅ ∈ V | |
7 | 5, 6 | eqeltrdi 2846 | . 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 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 Vcvv 3443 ∅c0 4280 {csn 4584 ℩cio 6443 |
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-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-sn 4585 df-pr 4587 df-uni 4864 df-iota 6445 |
This theorem is referenced by: iota4an 6475 fvex 6852 riotaex 7313 erov 8749 iunfictbso 10046 isf32lem9 10293 sumex 15564 prodex 15782 pcval 16708 grpidval 18508 fn0g 18510 gsumvalx 18523 psgnfn 19274 psgnval 19280 dchrptlem1 26596 lgsdchrval 26686 lgsdchr 26687 nosupno 27035 nosupdm 27036 nosupbday 27037 nosupfv 27038 nosupres 27039 nosupbnd1lem1 27040 noinfno 27050 noinfdm 27051 noinffv 27053 bnj1366 33310 bj-finsumval0 35723 ellimciota 43787 fourierdlem36 44316 eubrdm 45202 dfatafv2ex 45377 afv2ex 45378 funressndmafv2rn 45387 |
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