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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 4132 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵) | |
| 2 | 1 | con2i 139 | 1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 ∖ cdif 3948 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 |
| This theorem is referenced by: peano5 7915 extmptsuppeq 8213 undifixp 8974 ssfin4 10350 isf32lem3 10395 isf34lem4 10417 xrinfmss 13352 restntr 23190 cmpcld 23410 reconnlem2 24849 lebnumlem1 24993 i1fd 25716 ssdifidlprm 33486 hgt750lemd 34663 fmlasucdisj 35404 dfon2lem6 35789 onsucconni 36438 meaiininclem 46501 caragendifcl 46529 |
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