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| Mirrors > Home > MPE Home > Th. List > neldifsn | Structured version Visualization version GIF version | ||
| Description: The class 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| neldifsn | ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neirr 2949 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
| 2 | eldifsni 4790 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴) | |
| 3 | 1, 2 | mto 197 | 1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ≠ wne 2940 ∖ 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: neldifsnd 4793 fofinf1o 9372 dfac9 10177 1div0 11922 xrsupss 13351 hashgt23el 14463 fvsetsid 17205 islbs3 21157 islindf4 21858 ufinffr 23937 i1fd 25716 finsumvtxdg2sstep 29567 matunitlindflem1 37623 poimirlem25 37652 itg2addnclem 37678 itg2addnclem2 37679 prter2 38882 uspgrimprop 47873 |
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