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Mirrors > Home > ILE Home > Th. List > prnminu | Unicode version |
Description: An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.) |
Ref | Expression |
---|---|
prnminu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprnqu 7283 | . . . . 5 | |
2 | elinp 7275 | . . . . . . . 8 | |
3 | simpr1r 1039 | . . . . . . . 8 | |
4 | 2, 3 | sylbi 120 | . . . . . . 7 |
5 | eleq1 2200 | . . . . . . . . 9 | |
6 | breq2 3928 | . . . . . . . . . . 11 | |
7 | 6 | anbi1d 460 | . . . . . . . . . 10 |
8 | 7 | rexbidv 2436 | . . . . . . . . 9 |
9 | 5, 8 | bibi12d 234 | . . . . . . . 8 |
10 | 9 | rspcv 2780 | . . . . . . 7 |
11 | bi1 117 | . . . . . . 7 | |
12 | 4, 10, 11 | syl56 34 | . . . . . 6 |
13 | 12 | impd 252 | . . . . 5 |
14 | 1, 13 | mpcom 36 | . . . 4 |
15 | df-rex 2420 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | ltrelnq 7166 | . . . . . . . . 9 | |
18 | 17 | brel 4586 | . . . . . . . 8 |
19 | 18 | simpld 111 | . . . . . . 7 |
20 | 19 | pm4.71ri 389 | . . . . . 6 |
21 | 20 | anbi1i 453 | . . . . 5 |
22 | ancom 264 | . . . . 5 | |
23 | anass 398 | . . . . 5 | |
24 | 21, 22, 23 | 3bitr3i 209 | . . . 4 |
25 | 24 | exbii 1584 | . . 3 |
26 | 16, 25 | sylibr 133 | . 2 |
27 | df-rex 2420 | . 2 | |
28 | 26, 27 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 wss 3066 cop 3525 class class class wbr 3924 cnq 7081 cltq 7086 cnp 7092 |
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-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-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-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-iom 4500 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-qs 6428 df-ni 7105 df-nqqs 7149 df-ltnqqs 7154 df-inp 7267 |
This theorem is referenced by: genprndu 7323 nqpru 7353 1idpru 7392 ltsopr 7397 ltexprlemopu 7404 ltexprlemru 7413 addcanprlemu 7416 recexprlemloc 7432 recexprlem1ssu 7435 aptiprlemu 7441 |
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