Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > unirnioo | Unicode version |
Description: The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
unirnioo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioomax 8971 | . . . 4 | |
2 | ioof 8994 | . . . . . 6 | |
3 | ffn 5066 | . . . . . 6 | |
4 | 2, 3 | ax-mp 7 | . . . . 5 |
5 | mnfxr 8848 | . . . . 5 | |
6 | pnfxr 8846 | . . . . 5 | |
7 | fnovrn 5668 | . . . . 5 | |
8 | 4, 5, 6, 7 | mp3an 1268 | . . . 4 |
9 | 1, 8 | eqeltrri 2152 | . . 3 |
10 | elssuni 3629 | . . 3 | |
11 | 9, 10 | ax-mp 7 | . 2 |
12 | frn 5072 | . . . 4 | |
13 | 2, 12 | ax-mp 7 | . . 3 |
14 | sspwuni 3760 | . . 3 | |
15 | 13, 14 | mpbi 143 | . 2 |
16 | 11, 15 | eqssi 3015 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1284 wcel 1433 wss 2973 cpw 3382 cuni 3601 cxp 4361 crn 4364 wfn 4917 wf 4918 (class class class)co 5532 cr 6980 cpnf 7150 cmnf 7151 cxr 7152 cioo 8911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-ioo 8915 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |