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Mirrors > Home > MPE Home > Th. List > iotanul | Structured version Visualization version GIF version |
Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotanul | ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu6 2569 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | dfiota2 6497 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
3 | alnex 1784 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
4 | dfnul2 4326 | . . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧} | |
5 | equid 2016 | . . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧 | |
6 | 5 | tbt 370 | . . . . . . . . . . 11 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧)) |
7 | 6 | biimpi 215 | . . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧)) |
8 | 7 | con1bid 356 | . . . . . . . . 9 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
9 | 8 | alimi 1814 | . . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
10 | abbi 2801 | . . . . . . . 8 ⊢ (∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}) | |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}) |
12 | 4, 11 | eqtr2id 2786 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
13 | 3, 12 | sylbir 234 | . . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
14 | 13 | unieqd 4923 | . . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅) |
15 | uni0 4940 | . . . 4 ⊢ ∪ ∅ = ∅ | |
16 | 14, 15 | eqtrdi 2789 | . . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
17 | 2, 16 | eqtrid 2785 | . 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅) |
18 | 1, 17 | sylnbi 330 | 1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∃!weu 2563 {cab 2710 ∅c0 4323 ∪ cuni 4909 ℩cio 6494 |
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 |
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 3952 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 df-uni 4910 df-iota 6496 |
This theorem is referenced by: iotassuniOLD 6523 iotaexOLD 6524 iotan0 6534 dfiota4 6536 csbiota 6537 tz6.12-2 6880 dffv3 6888 csbriota 7381 riotaund 7405 isf32lem9 10356 grpidval 18580 0g0 18583 iota0ndef 45797 iotan0aiotaex 45849 |
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