| 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 2949 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
| 2 | eldifpr 4658 | . . 3 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ≠ 𝐴 ∧ 𝐴 ≠ 𝐵)) | |
| 3 | 2 | simp2bi 1147 | . 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 2108 ≠ wne 2940 ∖ cdif 3948 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |