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| Mirrors > Home > MPE Home > Th. List > neldifsnd | Structured version Visualization version GIF version | ||
| Description: The class 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.) | 
| Ref | Expression | 
|---|---|
| neldifsnd | ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | neldifsn 4792 | . 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) | |
| 2 | 1 | a1i 11 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 | 
| 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-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 | 
| This theorem is referenced by: difsnb 4806 fsnunf2 7206 fsumsplit1 15781 rpnnen2lem9 16258 fprodfvdvdsd 16371 ramub1lem1 17064 ramub1lem2 17065 prmdvdsprmo 17080 acsfiindd 18598 gsummgp0 20315 islindf4 21858 gsummatr01lem3 22663 nbgrnself 29376 omsmeas 34325 onint1 36450 bj-fvsnun2 37257 poimirlem30 37657 prtlem80 38862 aks6d1c5lem3 42138 gneispace0nelrn3 44155 supminfxr2 45480 fsumnncl 45587 hoidmv1lelem2 46607 hspmbllem1 46641 hspmbllem2 46642 fsumsplitsndif 47360 isubgr3stgrlem3 47935 mgpsumunsn 48277 | 
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