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Mirrors > Home > ILE Home > Th. List > genpcuu | Unicode version |
Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | |
genpelvl.2 | |
genpcuu.2 |
Ref | Expression |
---|---|
genpcuu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7286 | . . . . . . 7 | |
2 | 1 | brel 4639 | . . . . . 6 |
3 | 2 | simprd 113 | . . . . 5 |
4 | genpelvl.1 | . . . . . . . . 9 | |
5 | genpelvl.2 | . . . . . . . . 9 | |
6 | 4, 5 | genpelvu 7434 | . . . . . . . 8 |
7 | 6 | adantr 274 | . . . . . . 7 |
8 | breq1 3969 | . . . . . . . . . . . . 13 | |
9 | 8 | biimpd 143 | . . . . . . . . . . . 12 |
10 | genpcuu.2 | . . . . . . . . . . . 12 | |
11 | 9, 10 | sylan9r 408 | . . . . . . . . . . 11 |
12 | 11 | exp31 362 | . . . . . . . . . 10 |
13 | 12 | an4s 578 | . . . . . . . . 9 |
14 | 13 | impancom 258 | . . . . . . . 8 |
15 | 14 | rexlimdvv 2581 | . . . . . . 7 |
16 | 7, 15 | sylbid 149 | . . . . . 6 |
17 | 16 | ex 114 | . . . . 5 |
18 | 3, 17 | syl5 32 | . . . 4 |
19 | 18 | com34 83 | . . 3 |
20 | 19 | pm2.43d 50 | . 2 |
21 | 20 | com23 78 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wrex 2436 crab 2439 cop 3563 class class class wbr 3966 cfv 5171 (class class class)co 5825 cmpo 5827 c1st 6087 c2nd 6088 cnq 7201 cltq 7206 cnp 7212 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-qs 6487 df-ni 7225 df-nqqs 7269 df-ltnqqs 7274 df-inp 7387 |
This theorem is referenced by: genprndu 7443 |
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