Description: Given any set (the " y " in the statement), there exists a set not equal to it.
The same statement without disjoint variable condition is false, since we do not have E. x -. x = x . This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext , ax-sep , or ax-pow nor auxiliary logical axiom schemes ax-10 to ax-13 . See dtruALT for a shorter proof using more axioms, and dtruALT2 for a proof using ax-pow instead of ax-pr . (Contributed by NM, 7-Nov-2006) Avoid ax-13 . (Revised by BJ, 31-May-2019) Avoid ax-8 . (Revised by SN, 21-Sep-2023) Avoid ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024) Use ax-pr instead of ax-pow . (Revised by BTernaryTau, 3-Dec-2024) Extract this result from the proof of dtru . (Revised by BJ, 2-Jan-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | exneq | ⊢ ∃ 𝑥 ¬ 𝑥 = 𝑦 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exexneq | ⊢ ∃ 𝑧 ∃ 𝑤 ¬ 𝑧 = 𝑤 | |
| 2 | equeuclr | ⊢ ( 𝑤 = 𝑦 → ( 𝑧 = 𝑦 → 𝑧 = 𝑤 ) ) | |
| 3 | 2 | con3d | ⊢ ( 𝑤 = 𝑦 → ( ¬ 𝑧 = 𝑤 → ¬ 𝑧 = 𝑦 ) ) |
| 4 | ax7v1 | ⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 → 𝑧 = 𝑦 ) ) | |
| 5 | 4 | con3d | ⊢ ( 𝑥 = 𝑧 → ( ¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦 ) ) |
| 6 | 5 | spimevw | ⊢ ( ¬ 𝑧 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |
| 7 | 3 6 | syl6 | ⊢ ( 𝑤 = 𝑦 → ( ¬ 𝑧 = 𝑤 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) ) |
| 8 | ax7v1 | ⊢ ( 𝑥 = 𝑤 → ( 𝑥 = 𝑦 → 𝑤 = 𝑦 ) ) | |
| 9 | 8 | con3d | ⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦 ) ) |
| 10 | 9 | spimevw | ⊢ ( ¬ 𝑤 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |
| 11 | 10 | a1d | ⊢ ( ¬ 𝑤 = 𝑦 → ( ¬ 𝑧 = 𝑤 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) ) |
| 12 | 7 11 | pm2.61i | ⊢ ( ¬ 𝑧 = 𝑤 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |
| 13 | 12 | exlimivv | ⊢ ( ∃ 𝑧 ∃ 𝑤 ¬ 𝑧 = 𝑤 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) |
| 14 | 1 13 | ax-mp | ⊢ ∃ 𝑥 ¬ 𝑥 = 𝑦 |