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Mirrors > Home > MPE Home > Th. List > disjsn2 | Structured version Visualization version Unicode version |
Description: Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
disjsn2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4194 | . . . 4 | |
2 | 1 | eqcomd 2628 | . . 3 |
3 | 2 | necon3ai 2819 | . 2 |
4 | disjsn 4246 | . 2 | |
5 | 3, 4 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wceq 1483 wcel 1990 wne 2794 cin 3573 c0 3915 csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-nul 3916 df-sn 4178 |
This theorem is referenced by: disjpr2 4248 disjpr2OLD 4249 disjtpsn 4251 difprsn1 4330 diftpsn3OLD 4333 otsndisj 4979 xpsndisj 5557 funprg 5940 funprgOLD 5941 funtp 5945 funcnvpr 5950 f1oprg 6181 phplem1 8139 pm54.43 8826 pr2nelem 8827 f1oun2prg 13662 s3sndisj 13706 sumpr 14477 cshwsdisj 15805 setsfun0 15894 setscom 15903 xpsc0 16220 xpsc1 16221 dmdprdpr 18448 dprdpr 18449 ablfac1eulem 18471 cnfldfun 19758 m2detleib 20437 dishaus 21186 dissnlocfin 21332 xpstopnlem1 21612 perfectlem2 24955 prodpr 29572 esumpr 30128 esum2dlem 30154 prodfzo03 30681 onint1 32448 bj-disjsn01 32937 poimirlem26 33435 sumpair 39194 perfectALTVlem2 41631 gsumpr 42139 |
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