![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > exnel | Structured version Visualization version GIF version |
Description: There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.) |
Ref | Expression |
---|---|
exnel | ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 9588 | . 2 ⊢ ¬ 𝑦 ∈ 𝑦 | |
2 | 1 | nfth 1795 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦 |
3 | ax8 2104 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦)) | |
4 | 3 | con3d 152 | . . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦)) |
5 | 2, 4 | spime 2380 | . 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-13 2363 ax-ext 2695 ax-sep 5290 ax-pr 5418 ax-reg 9584 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-v 3468 df-un 3946 df-sn 4622 df-pr 4624 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |