Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prcunqu | Unicode version |
Description: An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
Ref | Expression |
---|---|
prcunqu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7173 | . . . . . 6 | |
2 | 1 | brel 4591 | . . . . 5 |
3 | 2 | simprd 113 | . . . 4 |
4 | 3 | adantl 275 | . . 3 |
5 | breq2 3933 | . . . . . . 7 | |
6 | eleq1 2202 | . . . . . . 7 | |
7 | 5, 6 | imbi12d 233 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | 1 | brel 4591 | . . . . . . . 8 |
10 | an42 576 | . . . . . . . . 9 | |
11 | breq1 3932 | . . . . . . . . . . . . . . . 16 | |
12 | eleq1 2202 | . . . . . . . . . . . . . . . 16 | |
13 | 11, 12 | anbi12d 464 | . . . . . . . . . . . . . . 15 |
14 | 13 | rspcev 2789 | . . . . . . . . . . . . . 14 |
15 | elinp 7282 | . . . . . . . . . . . . . . . 16 | |
16 | simpr1r 1039 | . . . . . . . . . . . . . . . 16 | |
17 | 15, 16 | sylbi 120 | . . . . . . . . . . . . . . 15 |
18 | 17 | r19.21bi 2520 | . . . . . . . . . . . . . 14 |
19 | 14, 18 | syl5ibrcom 156 | . . . . . . . . . . . . 13 |
20 | 19 | 3impb 1177 | . . . . . . . . . . . 12 |
21 | 20 | 3com12 1185 | . . . . . . . . . . 11 |
22 | 21 | 3expib 1184 | . . . . . . . . . 10 |
23 | 22 | impd 252 | . . . . . . . . 9 |
24 | 10, 23 | syl5bi 151 | . . . . . . . 8 |
25 | 9, 24 | mpand 425 | . . . . . . 7 |
26 | 25 | com12 30 | . . . . . 6 |
27 | 26 | ancoms 266 | . . . . 5 |
28 | 8, 27 | vtoclg 2746 | . . . 4 |
29 | 28 | impd 252 | . . 3 |
30 | 4, 29 | mpcom 36 | . 2 |
31 | 30 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 cop 3530 class class class wbr 3929 cnq 7088 cltq 7093 cnp 7099 |
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 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-qs 6435 df-ni 7112 df-nqqs 7156 df-ltnqqs 7161 df-inp 7274 |
This theorem is referenced by: prarloc 7311 prarloc2 7312 addnqprulem 7336 nqpru 7360 prmuloc2 7375 mulnqpru 7377 distrlem4pru 7393 1idpru 7399 ltexprlemm 7408 ltexprlemupu 7412 ltexprlemrl 7418 ltexprlemfu 7419 ltexprlemru 7420 aptiprlemu 7448 suplocexprlemdisj 7528 suplocexprlemub 7531 |
Copyright terms: Public domain | W3C validator |