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Mirrors > Home > ILE Home > Th. List > sneq | GIF version |
Description: Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sneq | ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2199 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)) | |
2 | 1 | abbidv 2307 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝐵}) |
3 | df-sn 3613 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
4 | df-sn 3613 | . 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵} | |
5 | 2, 3, 4 | 3eqtr4g 2247 | 1 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 {cab 2175 {csn 3607 |
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 6168 2ndvalg 6169 2ndval2 6182 fo1st 6183 fo2nd 6184 f1stres 6185 f2ndres 6186 mpomptsx 6223 dmmpossx 6225 fmpox 6226 brtpos2 6277 dftpos4 6289 tpostpos 6290 eceq1 6595 fvdiagfn 6720 mapsncnv 6722 elixpsn 6762 ixpsnf1o 6763 ensn1g 6824 en1 6826 xpsneng 6849 xpcomco 6853 xpassen 6857 xpdom2 6858 phplem3 6883 phplem3g 6885 fidifsnen 6899 xpfi 6959 pm54.43 7220 cc2lem 7296 cc2 7297 exp3val 10556 fsum2dlemstep 11477 fsumcnv 11480 fisumcom2 11481 fprod2dlemstep 11665 fprodcnv 11668 fprodcom2fi 11669 lssats2 13747 lspsneq0 13759 txswaphmeolem 14297 |
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