Mathbox for Thierry Arnoux |
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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 2952 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | eldifpr 4590 | . . 3 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ≠ 𝐴 ∧ 𝐴 ≠ 𝐵)) | |
3 | 2 | simp2bi 1148 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴 ≠ 𝐴) |
4 | 1, 3 | mto 200 | 1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2112 ≠ wne 2943 ∖ cdif 3880 {cpr 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-v 3425 df-dif 3886 df-un 3888 df-sn 4559 df-pr 4561 |
This theorem is referenced by: (None) |
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