![]() |
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 2948 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | eldifpr 4661 | . . 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 196 | 1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 ≠ wne 2939 ∖ cdif 3946 {cpr 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-v 3475 df-dif 3952 df-un 3954 df-sn 4630 df-pr 4632 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |