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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.) |
Ref | Expression |
---|---|
iotaex | ⊢ (℩𝑥𝜑) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 5862 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
2 | 1 | eqcomd 2628 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
3 | 2 | eximi 1762 | . . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑)) |
4 | df-eu 2474 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
5 | isset 3207 | . . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 281 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
7 | iotanul 5866 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
8 | 0ex 4790 | . . 3 ⊢ ∅ ∈ V | |
9 | 7, 8 | syl6eqel 2709 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
10 | 6, 9 | pm2.61i 176 | 1 ⊢ (℩𝑥𝜑) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 Vcvv 3200 ∅c0 3915 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 |
This theorem is referenced by: iota4an 5870 fvex 6201 riotaex 6615 erov 7844 iunfictbso 8937 isf32lem9 9183 sumex 14418 prodex 14637 pcval 15549 grpidval 17260 fn0g 17262 gsumvalx 17270 psgnfn 17921 psgnval 17927 dchrptlem1 24989 lgsdchrval 25079 lgsdchr 25080 bnj1366 30900 nosupno 31849 nosupdm 31850 nosupbday 31851 nosupfv 31852 nosupres 31853 nosupbnd1lem1 31854 bj-finsumval0 33147 ellimciota 39846 fourierdlem36 40360 |
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