Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prnmaxl | Unicode version |
Description: A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Ref | Expression |
---|---|
prnmaxl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprnql 7257 | . . . . 5 | |
2 | elinp 7250 | . . . . . . . 8 | |
3 | simpr1l 1023 | . . . . . . . 8 | |
4 | 2, 3 | sylbi 120 | . . . . . . 7 |
5 | eleq1 2180 | . . . . . . . . 9 | |
6 | breq1 3902 | . . . . . . . . . . 11 | |
7 | 6 | anbi1d 460 | . . . . . . . . . 10 |
8 | 7 | rexbidv 2415 | . . . . . . . . 9 |
9 | 5, 8 | bibi12d 234 | . . . . . . . 8 |
10 | 9 | rspcv 2759 | . . . . . . 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 2399 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | ltrelnq 7141 | . . . . . . . . 9 | |
18 | 17 | brel 4561 | . . . . . . . 8 |
19 | 18 | simprd 113 | . . . . . . 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 1569 | . . 3 |
26 | 16, 25 | sylibr 133 | . 2 |
27 | df-rex 2399 | . 2 | |
28 | 26, 27 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3a 947 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 wss 3041 cop 3500 class class class wbr 3899 cnq 7056 cltq 7061 cnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-qs 6403 df-ni 7080 df-nqqs 7124 df-ltnqqs 7129 df-inp 7242 |
This theorem is referenced by: prnmaddl 7266 genprndl 7297 nqprl 7327 1idprl 7366 ltsopr 7372 ltexprlemm 7376 ltexprlemopl 7377 recexprlemloc 7407 recexprlem1ssl 7409 aptiprleml 7415 caucvgprprlemopl 7473 suplocexprlemrl 7493 |
Copyright terms: Public domain | W3C validator |