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Mirrors > Home > ILE Home > Th. List > suplocexprlemlub | Unicode version |
Description: Lemma for suplocexpr 7533. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Ref | Expression |
---|---|
suplocexpr.m | |
suplocexpr.ub | |
suplocexpr.loc | |
suplocexpr.b |
Ref | Expression |
---|---|
suplocexprlemlub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 | |
2 | ltrelpr 7313 | . . . . . . . 8 | |
3 | 2 | brel 4591 | . . . . . . 7 |
4 | 3 | simpld 111 | . . . . . 6 |
5 | 4 | adantl 275 | . . . . 5 |
6 | 3 | simprd 113 | . . . . . 6 |
7 | 6 | adantl 275 | . . . . 5 |
8 | ltdfpr 7314 | . . . . 5 | |
9 | 5, 7, 8 | syl2anc 408 | . . . 4 |
10 | 1, 9 | mpbid 146 | . . 3 |
11 | simprrr 529 | . . . . . 6 | |
12 | suplocexpr.b | . . . . . . . . . 10 | |
13 | 12 | fveq2i 5424 | . . . . . . . . 9 |
14 | npex 7281 | . . . . . . . . . . . . 13 | |
15 | 14 | a1i 9 | . . . . . . . . . . . 12 |
16 | suplocexpr.m | . . . . . . . . . . . . 13 | |
17 | suplocexpr.ub | . . . . . . . . . . . . 13 | |
18 | suplocexpr.loc | . . . . . . . . . . . . 13 | |
19 | 16, 17, 18 | suplocexprlemss 7523 | . . . . . . . . . . . 12 |
20 | 15, 19 | ssexd 4068 | . . . . . . . . . . 11 |
21 | fo1st 6055 | . . . . . . . . . . . . 13 | |
22 | fofun 5346 | . . . . . . . . . . . . 13 | |
23 | 21, 22 | ax-mp 5 | . . . . . . . . . . . 12 |
24 | funimaexg 5207 | . . . . . . . . . . . 12 | |
25 | 23, 24 | mpan 420 | . . . . . . . . . . 11 |
26 | uniexg 4361 | . . . . . . . . . . 11 | |
27 | 20, 25, 26 | 3syl 17 | . . . . . . . . . 10 |
28 | nqex 7171 | . . . . . . . . . . 11 | |
29 | 28 | rabex 4072 | . . . . . . . . . 10 |
30 | op1stg 6048 | . . . . . . . . . 10 | |
31 | 27, 29, 30 | sylancl 409 | . . . . . . . . 9 |
32 | 13, 31 | syl5eq 2184 | . . . . . . . 8 |
33 | 32 | eleq2d 2209 | . . . . . . 7 |
34 | 33 | ad2antrr 479 | . . . . . 6 |
35 | 11, 34 | mpbid 146 | . . . . 5 |
36 | suplocexprlemell 7521 | . . . . 5 | |
37 | 35, 36 | sylib 121 | . . . 4 |
38 | simprl 520 | . . . . . . . . 9 | |
39 | 38 | ad2antrr 479 | . . . . . . . 8 |
40 | simprrl 528 | . . . . . . . . 9 | |
41 | 40 | ad2antrr 479 | . . . . . . . 8 |
42 | simpr 109 | . . . . . . . 8 | |
43 | rspe 2481 | . . . . . . . 8 | |
44 | 39, 41, 42, 43 | syl12anc 1214 | . . . . . . 7 |
45 | 4 | ad4antlr 486 | . . . . . . . 8 |
46 | 19 | ad4antr 485 | . . . . . . . . 9 |
47 | simplr 519 | . . . . . . . . 9 | |
48 | 46, 47 | sseldd 3098 | . . . . . . . 8 |
49 | ltdfpr 7314 | . . . . . . . 8 | |
50 | 45, 48, 49 | syl2anc 408 | . . . . . . 7 |
51 | 44, 50 | mpbird 166 | . . . . . 6 |
52 | 51 | ex 114 | . . . . 5 |
53 | 52 | reximdva 2534 | . . . 4 |
54 | 37, 53 | mpd 13 | . . 3 |
55 | 10, 54 | rexlimddv 2554 | . 2 |
56 | 55 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 crab 2420 cvv 2686 wss 3071 cop 3530 cuni 3736 cint 3771 class class class wbr 3929 cima 4542 wfun 5117 wfo 5121 cfv 5123 c1st 6036 c2nd 6037 cnq 7088 cltq 7093 cnp 7099 cltp 7103 |
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-1st 6038 df-qs 6435 df-ni 7112 df-nqqs 7156 df-inp 7274 df-iltp 7278 |
This theorem is referenced by: suplocexpr 7533 |
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