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Mirrors > Home > ILE Home > Th. List > opexg | Unicode version |
Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Ref | Expression |
---|---|
opexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3750 | . 2 | |
2 | elex 2732 | . . . . 5 | |
3 | snexg 4157 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | adantr 274 | . . 3 |
6 | elex 2732 | . . . 4 | |
7 | prexg 4183 | . . . 4 | |
8 | 2, 6, 7 | syl2an 287 | . . 3 |
9 | prexg 4183 | . . 3 | |
10 | 5, 8, 9 | syl2anc 409 | . 2 |
11 | 1, 10 | eqeltrd 2241 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cvv 2721 csn 3570 cpr 3571 cop 3573 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 |
This theorem is referenced by: opex 4201 otexg 4202 opeliunxp 4653 opbrop 4677 relsnopg 4702 opswapg 5084 elxp4 5085 elxp5 5086 resfunexg 5700 fliftel 5755 fliftel1 5756 oprabid 5865 ovexg 5867 eloprabga 5920 op1st 6106 op2nd 6107 ot1stg 6112 ot2ndg 6113 ot3rdgg 6114 elxp6 6129 mpofvex 6163 algrflem 6188 algrflemg 6189 mpoxopoveq 6199 brtposg 6213 tfr0dm 6281 tfrlemisucaccv 6284 tfrlemibxssdm 6286 tfrlemibfn 6287 tfrlemi14d 6292 tfr1onlemsucaccv 6300 tfr1onlembxssdm 6302 tfr1onlembfn 6303 tfr1onlemres 6308 tfrcllemsucaccv 6313 tfrcllembxssdm 6315 tfrcllembfn 6316 tfrcllemres 6321 fnfi 6893 djulclb 7011 inl11 7021 1stinl 7030 2ndinl 7031 1stinr 7032 2ndinr 7033 mulpipq2 7303 enq0breq 7368 addvalex 7776 peano2nnnn 7785 axcnre 7813 frec2uzrdg 10334 frecuzrdg0 10338 frecuzrdgg 10341 frecuzrdg0t 10347 zfz1isolem1 10739 eucalgval2 11964 crth 12135 phimullem 12136 ennnfonelemp1 12282 setsvala 12368 setsex 12369 setsfun 12372 setsfun0 12373 setsresg 12375 setscom 12377 strslfv 12381 setsslid 12387 strle1g 12427 1strbas 12436 2strbasg 12438 2stropg 12439 2strbas1g 12441 2strop1g 12442 rngbaseg 12453 rngplusgg 12454 rngmulrg 12455 srngbased 12460 srngplusgd 12461 srngmulrd 12462 srnginvld 12463 lmodbased 12471 lmodplusgd 12472 lmodscad 12473 lmodvscad 12474 ipsbased 12479 ipsaddgd 12480 ipsmulrd 12481 ipsscad 12482 ipsvscad 12483 ipsipd 12484 topgrpbasd 12489 topgrpplusgd 12490 topgrptsetd 12491 txlm 12826 |
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