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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 2574 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | dfiota2 6392 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
3 | alnex 1784 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
4 | dfnul2 4259 | . . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧} | |
5 | equid 2015 | . . . . . . . . . . . 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 | abbi1 2806 | . . . . . . . 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 2791 | . . . . . 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 4853 | . . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅) |
15 | uni0 4869 | . . . 4 ⊢ ∪ ∅ = ∅ | |
16 | 14, 15 | eqtrdi 2794 | . . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
17 | 2, 16 | eqtrid 2790 | . 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 1537 = wceq 1539 ∃wex 1782 ∃!weu 2568 {cab 2715 ∅c0 4256 ∪ cuni 4839 ℩cio 6389 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-v 3434 df-dif 3890 df-in 3894 df-ss 3904 df-nul 4257 df-sn 4562 df-uni 4840 df-iota 6391 |
This theorem is referenced by: iotassuni 6412 iotaex 6413 iotan0 6423 dfiota4 6425 csbiota 6426 tz6.12-2 6762 dffv3 6770 csbriota 7248 riotaund 7272 isf32lem9 10117 grpidval 18345 0g0 18348 iota0ndef 44533 iotan0aiotaex 44585 |
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