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Mirrors > Home > ILE Home > Th. List > ltxr | Unicode version |
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ltxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq12 3904 | . . . . 5 | |
2 | df-3an 949 | . . . . . 6 | |
3 | 2 | opabbii 3965 | . . . . 5 |
4 | 1, 3 | brab2ga 4584 | . . . 4 |
5 | 4 | a1i 9 | . . 3 |
6 | brun 3949 | . . . 4 | |
7 | brxp 4540 | . . . . . . 7 | |
8 | elun 3187 | . . . . . . . . . . 11 | |
9 | orcom 702 | . . . . . . . . . . 11 | |
10 | 8, 9 | bitri 183 | . . . . . . . . . 10 |
11 | elsng 3512 | . . . . . . . . . . 11 | |
12 | 11 | orbi1d 765 | . . . . . . . . . 10 |
13 | 10, 12 | syl5bb 191 | . . . . . . . . 9 |
14 | elsng 3512 | . . . . . . . . 9 | |
15 | 13, 14 | bi2anan9 580 | . . . . . . . 8 |
16 | andir 793 | . . . . . . . 8 | |
17 | 15, 16 | syl6bb 195 | . . . . . . 7 |
18 | 7, 17 | syl5bb 191 | . . . . . 6 |
19 | brxp 4540 | . . . . . . 7 | |
20 | 11 | anbi1d 460 | . . . . . . . 8 |
21 | 20 | adantr 274 | . . . . . . 7 |
22 | 19, 21 | syl5bb 191 | . . . . . 6 |
23 | 18, 22 | orbi12d 767 | . . . . 5 |
24 | orass 741 | . . . . 5 | |
25 | 23, 24 | syl6bb 195 | . . . 4 |
26 | 6, 25 | syl5bb 191 | . . 3 |
27 | 5, 26 | orbi12d 767 | . 2 |
28 | df-ltxr 7773 | . . . 4 | |
29 | 28 | breqi 3905 | . . 3 |
30 | brun 3949 | . . 3 | |
31 | 29, 30 | bitri 183 | . 2 |
32 | orass 741 | . 2 | |
33 | 27, 31, 32 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 w3a 947 wceq 1316 wcel 1465 cun 3039 csn 3497 class class class wbr 3899 copab 3958 cxp 4507 cr 7587 cltrr 7592 cpnf 7765 cmnf 7766 cxr 7767 clt 7768 |
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-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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
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-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-ltxr 7773 |
This theorem is referenced by: xrltnr 9534 ltpnf 9535 mnflt 9537 mnfltpnf 9539 pnfnlt 9541 nltmnf 9542 |
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