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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 2572 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | dfiota2 6449 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
3 | alnex 1783 | . . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
4 | dfnul2 4285 | . . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧} | |
5 | equid 2015 | . . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧 | |
6 | 5 | tbt 369 | . . . . . . . . . . 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 355 | . . . . . . . . 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 2804 | . . . . . . . 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 2789 | . . . . . 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 4879 | . . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅) |
15 | uni0 4896 | . . . 4 ⊢ ∪ ∅ = ∅ | |
16 | 14, 15 | eqtrdi 2792 | . . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅) |
17 | 2, 16 | eqtrid 2788 | . 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅) |
18 | 1, 17 | sylnbi 329 | 1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1539 = wceq 1541 ∃wex 1781 ∃!weu 2566 {cab 2713 ∅c0 4282 ∪ cuni 4865 ℩cio 6446 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-v 3447 df-dif 3913 df-in 3917 df-ss 3927 df-nul 4283 df-sn 4587 df-uni 4866 df-iota 6448 |
This theorem is referenced by: iotassuniOLD 6475 iotaexOLD 6476 iotan0 6486 dfiota4 6488 csbiota 6489 tz6.12-2 6830 dffv3 6838 csbriota 7328 riotaund 7352 isf32lem9 10296 grpidval 18515 0g0 18518 iota0ndef 45244 iotan0aiotaex 45296 |
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