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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 4055 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵) | |
2 | 1 | con2i 141 | 1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2111 ∖ cdif 3878 |
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-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 |
This theorem is referenced by: peano5 7585 extmptsuppeq 7837 undifixp 8481 ssfin4 9721 isf32lem3 9766 isf34lem4 9788 xrinfmss 12691 restntr 21787 cmpcld 22007 reconnlem2 23432 lebnumlem1 23566 i1fd 24285 hgt750lemd 32029 fmlasucdisj 32759 dfon2lem6 33146 onsucconni 33898 meaiininclem 43125 caragendifcl 43153 |
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