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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 6003 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
2 | 1 | eqcomd 2777 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
3 | 2 | eximi 1910 | . . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑)) |
4 | df-eu 2622 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
5 | isset 3359 | . . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 281 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
7 | iotanul 6007 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
8 | 0ex 4924 | . . 3 ⊢ ∅ ∈ V | |
9 | 7, 8 | syl6eqel 2858 | . 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 1629 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ∃!weu 2618 Vcvv 3351 ∅c0 4063 ℩cio 5990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-nul 4923 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-sn 4317 df-pr 4319 df-uni 4575 df-iota 5992 |
This theorem is referenced by: iota4an 6011 fvex 6340 riotaex 6756 erov 7995 iunfictbso 9135 isf32lem9 9383 sumex 14619 prodex 14837 pcval 15749 grpidval 17461 fn0g 17463 gsumvalx 17471 psgnfn 18121 psgnval 18127 dchrptlem1 25203 lgsdchrval 25293 lgsdchr 25294 bnj1366 31231 nosupno 32179 nosupdm 32180 nosupbday 32181 nosupfv 32182 nosupres 32183 nosupbnd1lem1 32184 bj-finsumval0 33477 ellimciota 40357 fourierdlem36 40870 |
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