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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > neldifpr2 | Structured version Visualization version GIF version |
Description: The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
neldifpr2 | ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neirr 2943 | . 2 ⊢ ¬ 𝐵 ≠ 𝐵 | |
2 | eldifpr 4655 | . . 3 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵 ∈ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐵)) | |
3 | 2 | simp3bi 1144 | . 2 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵 ≠ 𝐵) |
4 | 1, 3 | mto 196 | 1 ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 {cpr 4625 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-un 3948 df-sn 4624 df-pr 4626 |
This theorem is referenced by: (None) |
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