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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 3025 | . 2 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | eldifsni 4716 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴) | |
3 | 1, 2 | mto 199 | 1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 {csn 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3497 df-dif 3939 df-sn 4562 |
This theorem is referenced by: neldifsnd 4720 fofinf1o 8793 dfac9 9556 xrsupss 12696 hashgt23el 13779 fvsetsid 16509 islbs3 19921 islindf4 20976 ufinffr 22531 i1fd 24276 finsumvtxdg2sstep 27325 matunitlindflem1 34882 poimirlem25 34911 itg2addnclem 34937 itg2addnclem2 34938 prter2 36011 |
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