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Mirrors > Home > MPE Home > Th. List > 0in | Structured version Visualization version GIF version |
Description: The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
0in | ⊢ (∅ ∩ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3767 | . 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅) | |
2 | in0 3920 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
3 | 1, 2 | eqtri 2632 | 1 ⊢ (∅ ∩ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∩ cin 3539 ∅c0 3874 |
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-v 3175 df-dif 3543 df-in 3547 df-nul 3875 |
This theorem is referenced by: pred0 5627 fnsuppeq0 7210 setsfun 15725 setsfun0 15726 indistopon 20615 fctop 20618 cctop 20620 restsn 20784 filcon 21497 chtdif 24684 ppidif 24689 ppi1 24690 cht1 24691 ofpreima2 28849 ordtconlem1 29298 measvuni 29604 measinb 29611 cndprobnul 29826 ballotlemfp1 29880 ballotlemgun 29913 mrsubvrs 30673 mblfinlem2 32617 subsalsal 39253 nnfoctbdjlem 39348 |
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