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Mirrors > Home > MPE Home > Th. List > dtru | Structured version Visualization version GIF version |
Description: At least two sets exist
(or in terms of first-order logic, the universe
of discourse has two or more objects). Note that we may not substitute
the same variable for both 𝑥 and 𝑦 (as indicated by the
distinct
variable requirement), for otherwise we would contradict stdpc6 2132.
This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2803 or ax-sep 5005. See dtruALT 5087 for a shorter proof using these axioms. The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.) |
Ref | Expression |
---|---|
dtru | ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 5069 | . . . 4 ⊢ ∃𝑤 𝑥 ∈ 𝑤 | |
2 | ax-nul 5013 | . . . . 5 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧 | |
3 | sp 2224 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧) | |
4 | 2, 3 | eximii 1935 | . . . 4 ⊢ ∃𝑧 ¬ 𝑥 ∈ 𝑧 |
5 | exdistrv 2054 | . . . 4 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤 𝑥 ∈ 𝑤 ∧ ∃𝑧 ¬ 𝑥 ∈ 𝑧)) | |
6 | 1, 4, 5 | mpbir2an 702 | . . 3 ⊢ ∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) |
7 | ax9 2177 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧)) | |
8 | 7 | com12 32 | . . . . 5 ⊢ (𝑥 ∈ 𝑤 → (𝑤 = 𝑧 → 𝑥 ∈ 𝑧)) |
9 | 8 | con3dimp 399 | . . . 4 ⊢ ((𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧) |
10 | 9 | 2eximi 1934 | . . 3 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧) |
11 | equequ2 2130 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑤 = 𝑦)) | |
12 | 11 | notbid 310 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦)) |
13 | ax7 2120 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦)) | |
14 | 13 | con3d 150 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦)) |
15 | 14 | spimev 2412 | . . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦) |
16 | 12, 15 | syl6bi 245 | . . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)) |
17 | ax7 2120 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦)) | |
18 | 17 | con3d 150 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦)) |
19 | 18 | spimev 2412 | . . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦) |
20 | 19 | a1d 25 | . . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)) |
21 | 16, 20 | pm2.61i 177 | . . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦) |
22 | 21 | exlimivv 2031 | . . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦) |
23 | 6, 10, 22 | mp2b 10 | . 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 |
24 | exnal 1925 | . 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | |
25 | 23, 24 | mpbi 222 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∀wal 1654 ∃wex 1878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-12 2220 ax-13 2389 ax-nul 5013 ax-pow 5065 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1660 df-ex 1879 df-nf 1883 |
This theorem is referenced by: dtrucor 5071 dvdemo1 5073 nfnid 5075 axc16b 5088 eunex 5089 eunexOLD 5090 brprcneu 6425 zfcndpow 9753 |
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