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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 4554 | . 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
4 | 3 | nelir 3094 | 1 ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2987 ∉ wnel 3091 {cpr 4527 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-nel 3092 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 |
This theorem is referenced by: vdegp1ai 27326 |
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