**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 .

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

Ref | Expression | ||
---|---|---|---|

Assertion | dtru | ⊢ ¬ ∀ 𝑥 𝑥 = 𝑦 |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | el | ⊢ ∃ 𝑤 𝑥 ∈ 𝑤 | |

2 | ax-nul | ⊢ ∃ 𝑧 ∀ 𝑥 ¬ 𝑥 ∈ 𝑧 | |

3 | sp | ⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧 ) | |

4 | 2 3 | eximii | ⊢ ∃ 𝑧 ¬ 𝑥 ∈ 𝑧 |

5 | exdistrv | ⊢ ( ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) ↔ ( ∃ 𝑤 𝑥 ∈ 𝑤 ∧ ∃ 𝑧 ¬ 𝑥 ∈ 𝑧 ) ) | |

6 | 1 4 5 | mpbir2an | ⊢ ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) |

7 | ax9v2 | ⊢ ( 𝑤 = 𝑧 → ( 𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧 ) ) | |

8 | 7 | com12 | ⊢ ( 𝑥 ∈ 𝑤 → ( 𝑤 = 𝑧 → 𝑥 ∈ 𝑧 ) ) |

9 | 8 | con3dimp | ⊢ ( ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) → ¬ 𝑤 = 𝑧 ) |

10 | 9 | 2eximi | ⊢ ( ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) → ∃ 𝑤 ∃ 𝑧 ¬ 𝑤 = 𝑧 ) |

11 | equequ2 | ⊢ ( 𝑧 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑤 = 𝑦 ) ) | |

12 | 11 | notbid | ⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦 ) ) |

13 | nfv | ⊢ Ⅎ 𝑥 ¬ 𝑤 = 𝑦 | |

14 | ax7v1 | ⊢ ( 𝑥 = 𝑤 → ( 𝑥 = 𝑦 → 𝑤 = 𝑦 ) ) | |

15 | 14 | con3d | ⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦 ) ) |

16 | 13 15 | spimefv | ⊢ ( ¬ 𝑤 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |

17 | 12 16 | syl6bi | ⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) ) |

18 | nfv | ⊢ Ⅎ 𝑥 ¬ 𝑧 = 𝑦 | |

19 | ax7v1 | ⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 → 𝑧 = 𝑦 ) ) | |

20 | 19 | con3d | ⊢ ( 𝑥 = 𝑧 → ( ¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦 ) ) |

21 | 18 20 | spimefv | ⊢ ( ¬ 𝑧 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |

22 | 21 | a1d | ⊢ ( ¬ 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) ) |

23 | 17 22 | pm2.61i | ⊢ ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |

24 | 23 | exlimivv | ⊢ ( ∃ 𝑤 ∃ 𝑧 ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |

25 | 6 10 24 | mp2b | ⊢ ∃ 𝑥 ¬ 𝑥 = 𝑦 |

26 | exnal | ⊢ ( ∃ 𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀ 𝑥 𝑥 = 𝑦 ) | |

27 | 25 26 | mpbi | ⊢ ¬ ∀ 𝑥 𝑥 = 𝑦 |