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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 4596 | . 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} |
4 | 3 | nelir 3128 | 1 ⊢ 𝐴 ∉ {𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3018 ∉ wnel 3125 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 |
This theorem is referenced by: vdegp1ai 27320 |
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