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Mirrors > Home > ILE Home > Th. List > toponmax | GIF version |
Description: The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toponmax | β’ (π½ β (TopOnβπ΅) β π΅ β π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 13518 | . 2 β’ (π½ β (TopOnβπ΅) β π΅ = βͺ π½) | |
2 | topontop 13517 | . . 3 β’ (π½ β (TopOnβπ΅) β π½ β Top) | |
3 | eqid 2177 | . . . 4 β’ βͺ π½ = βͺ π½ | |
4 | 3 | topopn 13511 | . . 3 β’ (π½ β Top β βͺ π½ β π½) |
5 | 2, 4 | syl 14 | . 2 β’ (π½ β (TopOnβπ΅) β βͺ π½ β π½) |
6 | 1, 5 | eqeltrd 2254 | 1 β’ (π½ β (TopOnβπ΅) β π΅ β π½) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wcel 2148 βͺ cuni 3810 βcfv 5217 Topctop 13500 TopOnctopon 13513 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-top 13501 df-topon 13514 |
This theorem is referenced by: topgele 13532 eltpsg 13543 resttopon 13674 lmfval 13695 cnfval 13697 cnpfval 13698 iscn 13700 cnpval 13701 iscnp 13702 lmbrf 13718 cnconst2 13736 cnrest2 13739 cndis 13744 cnpdis 13745 lmfss 13747 lmres 13751 lmff 13752 tx1cn 13772 tx2cn 13773 txlm 13782 cnmpt2res 13800 mopnm 13951 isxms2 13955 |
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