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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 9724 | . . . 4 | |
2 | ioof 9747 | . . . . . 6 | |
3 | ffn 5267 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | mnfxr 7815 | . . . . 5 | |
6 | pnfxr 7811 | . . . . 5 | |
7 | fnovrn 5911 | . . . . 5 | |
8 | 4, 5, 6, 7 | mp3an 1315 | . . . 4 |
9 | 1, 8 | eqeltrri 2211 | . . 3 |
10 | elssuni 3759 | . . 3 | |
11 | 9, 10 | ax-mp 5 | . 2 |
12 | frn 5276 | . . . 4 | |
13 | 2, 12 | ax-mp 5 | . . 3 |
14 | sspwuni 3892 | . . 3 | |
15 | 13, 14 | mpbi 144 | . 2 |
16 | 11, 15 | eqssi 3108 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 wss 3066 cpw 3505 cuni 3731 cxp 4532 crn 4535 wfn 5113 wf 5114 (class class class)co 5767 cr 7612 cpnf 7790 cmnf 7791 cxr 7792 cioo 9664 |
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-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-nel 2402 df-ral 2419 df-rex 2420 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-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 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-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-ioo 9668 |
This theorem is referenced by: uniretop 12683 tgioo 12704 |
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