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Theorem dtru 5259
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 2036.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2796 or ax-sep 5190. See dtruALT 5277 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.) Avoid ax-13 2392. (Revised by Gino Giotto, 5-Sep-2023.)

Assertion
Ref Expression
dtru ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtru
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el 5258 . . . 4 𝑤 𝑥𝑤
2 ax-nul 5197 . . . . 5 𝑧𝑥 ¬ 𝑥𝑧
3 sp 2184 . . . . 5 (∀𝑥 ¬ 𝑥𝑧 → ¬ 𝑥𝑧)
42, 3eximii 1838 . . . 4 𝑧 ¬ 𝑥𝑧
5 exdistrv 1957 . . . 4 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) ↔ (∃𝑤 𝑥𝑤 ∧ ∃𝑧 ¬ 𝑥𝑧))
61, 4, 5mpbir2an 710 . . 3 𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧)
7 ax9v2 2128 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
87com12 32 . . . . 5 (𝑥𝑤 → (𝑤 = 𝑧𝑥𝑧))
98con3dimp 412 . . . 4 ((𝑥𝑤 ∧ ¬ 𝑥𝑧) → ¬ 𝑤 = 𝑧)
1092eximi 1837 . . 3 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) → ∃𝑤𝑧 ¬ 𝑤 = 𝑧)
11 equequ2 2034 . . . . . . 7 (𝑧 = 𝑦 → (𝑤 = 𝑧𝑤 = 𝑦))
1211notbid 321 . . . . . 6 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
13 nfv 1916 . . . . . . 7 𝑥 ¬ 𝑤 = 𝑦
14 ax7v1 2018 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
1514con3d 155 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
1613, 15spimefv 2200 . . . . . 6 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
1712, 16syl6bi 256 . . . . 5 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
18 nfv 1916 . . . . . . 7 𝑥 ¬ 𝑧 = 𝑦
19 ax7v1 2018 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
2019con3d 155 . . . . . . 7 (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
2118, 20spimefv 2200 . . . . . 6 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
2221a1d 25 . . . . 5 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
2317, 22pm2.61i 185 . . . 4 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
2423exlimivv 1934 . . 3 (∃𝑤𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
256, 10, 24mp2b 10 . 2 𝑥 ¬ 𝑥 = 𝑦
26 exnal 1828 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2725, 26mpbi 233 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-nul 5197  ax-pow 5254
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by:  dtrucor  5260  dvdemo1  5262  nfnid  5264  axc16b  5278  eunex  5279  brprcneu  6651  zfcndpow  10032
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