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Mirrors > Home > ILE Home > Th. List > preqlu | Unicode version |
Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
preqlu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | npsspw 6661 | . . . . 5 | |
2 | 1 | sseli 2995 | . . . 4 |
3 | 1st2nd2 5821 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 1 | sseli 2995 | . . . 4 |
6 | 1st2nd2 5821 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | eqeqan12d 2096 | . 2 |
9 | xp1st 5812 | . . . . 5 | |
10 | 2, 9 | syl 14 | . . . 4 |
11 | xp2nd 5813 | . . . . 5 | |
12 | 2, 11 | syl 14 | . . . 4 |
13 | opthg 3993 | . . . 4 | |
14 | 10, 12, 13 | syl2anc 403 | . . 3 |
15 | 14 | adantr 270 | . 2 |
16 | 8, 15 | bitrd 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cpw 3382 cop 3401 cxp 4361 cfv 4922 c1st 5785 c2nd 5786 cnq 6470 cnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fv 4930 df-1st 5787 df-2nd 5788 df-inp 6656 |
This theorem is referenced by: genpassg 6716 addnqpr 6751 mulnqpr 6767 distrprg 6778 1idpr 6782 ltexpri 6803 addcanprg 6806 recexprlemex 6827 aptipr 6831 |
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