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Mirrors > Home > MPE Home > Th. List > elndif | Structured version Visualization version GIF version |
Description: A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
elndif | ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifn 3695 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵) | |
2 | 1 | con2i 133 | 1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ∖ cdif 3537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 |
This theorem is referenced by: peano5 6981 extmptsuppeq 7206 undifixp 7830 ssfin4 9015 isf32lem3 9060 isf34lem4 9082 xrinfmss 12012 restntr 20796 cmpcld 21015 reconnlem2 22438 lebnumlem1 22568 i1fd 23254 dfon2lem6 30937 onsucconi 31606 meaiininclem 39376 caragendifcl 39404 |
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