Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > topontopn | Unicode version |
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tsettps.a | |
tsettps.j | TopSet |
Ref | Expression |
---|---|
topontopn | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12170 | . . 3 TopOn | |
2 | tsetslid 12098 | . . . . . 6 TopSet Slot TopSet TopSet | |
3 | 2 | slotslfn 11974 | . . . . 5 TopSet |
4 | fnrel 5216 | . . . . 5 TopSet TopSet | |
5 | 3, 4 | ax-mp 5 | . . . 4 TopSet |
6 | 0opn 12162 | . . . . 5 | |
7 | tsettps.j | . . . . 5 TopSet | |
8 | 6, 7 | eleqtrdi 2230 | . . . 4 TopSet |
9 | relelfvdm 5446 | . . . 4 TopSet TopSet TopSet | |
10 | 5, 8, 9 | sylancr 410 | . . 3 TopSet |
11 | 1, 10 | syl 14 | . 2 TopOn TopSet |
12 | toponuni 12171 | . . . 4 TopOn | |
13 | eqimss2 3147 | . . . 4 | |
14 | 12, 13 | syl 14 | . . 3 TopOn |
15 | sspwuni 3892 | . . 3 | |
16 | 14, 15 | sylibr 133 | . 2 TopOn |
17 | tsettps.a | . . 3 | |
18 | 17, 7 | topnidg 12122 | . 2 TopSet |
19 | 11, 16, 18 | syl2anc 408 | 1 TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cvv 2681 wss 3066 c0 3358 cpw 3505 cuni 3731 cdm 4534 wrel 4539 wfn 5113 cfv 5118 cbs 11948 TopSetcts 12016 ctopn 12110 ctop 12153 TopOnctopon 12166 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 df-ndx 11951 df-slot 11952 df-base 11954 df-tset 12029 df-rest 12111 df-topn 12112 df-top 12154 df-topon 12167 |
This theorem is referenced by: tsettps 12194 |
Copyright terms: Public domain | W3C validator |