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Mirrors > Home > ILE Home > Th. List > sneq | Unicode version |
Description: Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sneq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2199 |
. . 3
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2 | 1 | abbidv 2307 |
. 2
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3 | df-sn 3613 |
. 2
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4 | df-sn 3613 |
. 2
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5 | 2, 3, 4 | 3eqtr4g 2247 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-sn 3613 |
This theorem is referenced by: sneqi 3619 sneqd 3620 euabsn 3677 absneu 3679 preq1 3684 tpeq3 3695 snssgOLD 3743 sneqrg 3777 sneqbg 3778 opeq1 3793 unisng 3841 exmidsssn 4220 exmidsssnc 4221 suceq 4420 snnex 4466 opeliunxp 4699 relop 4795 elimasng 5014 dmsnsnsng 5124 elxp4 5134 elxp5 5135 iotajust 5195 fconstg 5431 f1osng 5521 nfvres 5568 fsng 5710 fnressn 5723 fressnfv 5724 funfvima3 5771 isoselem 5842 1stvalg 6167 2ndvalg 6168 2ndval2 6181 fo1st 6182 fo2nd 6183 f1stres 6184 f2ndres 6185 mpomptsx 6222 dmmpossx 6224 fmpox 6225 brtpos2 6276 dftpos4 6288 tpostpos 6289 eceq1 6594 fvdiagfn 6719 mapsncnv 6721 elixpsn 6761 ixpsnf1o 6762 ensn1g 6823 en1 6825 xpsneng 6848 xpcomco 6852 xpassen 6856 xpdom2 6857 phplem3 6882 phplem3g 6884 fidifsnen 6898 xpfi 6958 pm54.43 7219 cc2lem 7295 cc2 7296 exp3val 10553 fsum2dlemstep 11474 fsumcnv 11477 fisumcom2 11478 fprod2dlemstep 11662 fprodcnv 11665 fprodcom2fi 11666 lssats2 13730 lspsneq0 13742 txswaphmeolem 14277 |
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