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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 4544 | . 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
4 | 3 | nelir 3056 | 1 ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2949 ∉ wnel 3053 {cpr 4517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ne 2950 df-nel 3054 df-v 3409 df-un 3859 df-sn 4516 df-pr 4518 |
This theorem is referenced by: vdegp1ai 27410 |
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