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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 2791 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | eldifsni 4261 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴) | |
3 | 1, 2 | mto 187 | 1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 |
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-ne 2782 df-v 3175 df-dif 3543 df-sn 4126 |
This theorem is referenced by: neldifsnd 4263 fofinf1o 8126 dfac9 8841 xrsupss 12011 fvsetsid 15722 islbs3 18976 islindf4 19996 ufinffr 21543 i1fd 23254 nbgranself2 25965 matunitlindflem1 32575 poimirlem25 32604 itg2addnclem 32631 itg2addnclem2 32632 prter2 33184 |
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