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Mirrors > Home > ILE Home > Th. List > snexg | Unicode version |
Description: A singleton whose element exists is a set. The case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.) |
Ref | Expression |
---|---|
snexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4099 | . 2 | |
2 | snsspw 3686 | . . 3 | |
3 | ssexg 4062 | . . 3 | |
4 | 2, 3 | mpan 420 | . 2 |
5 | 1, 4 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 cvv 2681 wss 3066 cpw 3505 csn 3522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 |
This theorem is referenced by: snex 4104 notnotsnex 4106 exmidsssnc 4121 snelpwi 4129 opexg 4145 opm 4151 tpexg 4360 op1stbg 4395 sucexb 4408 elxp4 5021 elxp5 5022 opabex3d 6012 opabex3 6013 1stvalg 6033 2ndvalg 6034 mpoexxg 6101 cnvf1o 6115 brtpos2 6141 tfr0dm 6212 tfrlemisucaccv 6215 tfrlemibxssdm 6217 tfrlemibfn 6218 tfr1onlemsucaccv 6231 tfr1onlembxssdm 6233 tfr1onlembfn 6234 tfrcllemsucaccv 6244 tfrcllembxssdm 6246 tfrcllembfn 6247 fvdiagfn 6580 ixpsnf1o 6623 mapsnf1o 6624 xpsnen2g 6716 zfz1isolem1 10576 climconst2 11053 ennnfonelemp1 11908 setsvalg 11978 setsex 11980 setsslid 11998 strle1g 12038 1strbas 12047 |
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