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Mirrors > Home > MPE Home > Th. List > prneli | Structured version Visualization version GIF version |
Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
prneli.1 | ⊢ 𝐴 ≠ 𝐵 |
prneli.2 | ⊢ 𝐴 ≠ 𝐶 |
Ref | Expression |
---|---|
prneli | ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prneli.1 | . . 3 ⊢ 𝐴 ≠ 𝐵 | |
2 | prneli.2 | . . 3 ⊢ 𝐴 ≠ 𝐶 | |
3 | 1, 2 | nelpri 4613 | . 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
4 | 3 | nelir 3050 | 1 ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2941 ∉ wnel 3047 {cpr 4586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-nel 3048 df-v 3445 df-un 3913 df-sn 4585 df-pr 4587 |
This theorem is referenced by: vdegp1ai 28370 |
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