![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > neldifpr1 | Structured version Visualization version GIF version |
Description: The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
neldifpr1 | ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neirr 2946 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | eldifpr 4662 | . . 3 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ≠ 𝐴 ∧ 𝐴 ≠ 𝐵)) | |
3 | 2 | simp2bi 1145 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴 ≠ 𝐴) |
4 | 1, 3 | mto 197 | 1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {cpr 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-un 3967 df-sn 4631 df-pr 4633 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |