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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 2634 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | dfiota2 6284 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
3 | alnex 1783 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
4 | dfnul2 4245 | . . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧} | |
5 | equid 2019 | . . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧 | |
6 | 5 | tbt 373 | . . . . . . . . . . 11 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧)) |
7 | 6 | biimpi 219 | . . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧)) |
8 | 7 | con1bid 359 | . . . . . . . . 9 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
9 | 8 | alimi 1813 | . . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
10 | abbi1 2861 | . . . . . . . 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 | syl5req 2846 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
13 | 3, 12 | sylbir 238 | . . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
14 | 13 | unieqd 4814 | . . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅) |
15 | uni0 4828 | . . . 4 ⊢ ∪ ∅ = ∅ | |
16 | 14, 15 | eqtrdi 2849 | . . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
17 | 2, 16 | syl5eq 2845 | . 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅) |
18 | 1, 17 | sylnbi 333 | 1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 ∃!weu 2628 {cab 2776 ∅c0 4243 ∪ cuni 4800 ℩cio 6281 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-uni 4801 df-iota 6283 |
This theorem is referenced by: iotassuni 6303 iotaex 6304 iotan0 6314 dfiota4 6316 csbiota 6317 tz6.12-2 6635 dffv3 6641 csbriota 7108 riotaund 7132 isf32lem9 9772 grpidval 17863 0g0 17866 iota0ndef 43631 iotan0aiotaex 43648 |
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