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Mirrors > Home > ILE Home > Th. List > xrltnr | Unicode version |
Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
xrltnr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9518 | . 2 | |
2 | ltnr 7809 | . . 3 | |
3 | pnfnre 7775 | . . . . . . . . . 10 | |
4 | 3 | neli 2382 | . . . . . . . . 9 |
5 | 4 | intnan 899 | . . . . . . . 8 |
6 | 5 | intnanr 900 | . . . . . . 7 |
7 | pnfnemnf 7788 | . . . . . . . . 9 | |
8 | 7 | neii 2287 | . . . . . . . 8 |
9 | 8 | intnanr 900 | . . . . . . 7 |
10 | 6, 9 | pm3.2ni 787 | . . . . . 6 |
11 | 4 | intnanr 900 | . . . . . . 7 |
12 | 4 | intnan 899 | . . . . . . 7 |
13 | 11, 12 | pm3.2ni 787 | . . . . . 6 |
14 | 10, 13 | pm3.2ni 787 | . . . . 5 |
15 | pnfxr 7786 | . . . . . 6 | |
16 | ltxr 9517 | . . . . . 6 | |
17 | 15, 15, 16 | mp2an 422 | . . . . 5 |
18 | 14, 17 | mtbir 645 | . . . 4 |
19 | breq12 3904 | . . . . 5 | |
20 | 19 | anidms 394 | . . . 4 |
21 | 18, 20 | mtbiri 649 | . . 3 |
22 | mnfnre 7776 | . . . . . . . . . 10 | |
23 | 22 | neli 2382 | . . . . . . . . 9 |
24 | 23 | intnan 899 | . . . . . . . 8 |
25 | 24 | intnanr 900 | . . . . . . 7 |
26 | 7 | nesymi 2331 | . . . . . . . 8 |
27 | 26 | intnan 899 | . . . . . . 7 |
28 | 25, 27 | pm3.2ni 787 | . . . . . 6 |
29 | 23 | intnanr 900 | . . . . . . 7 |
30 | 23 | intnan 899 | . . . . . . 7 |
31 | 29, 30 | pm3.2ni 787 | . . . . . 6 |
32 | 28, 31 | pm3.2ni 787 | . . . . 5 |
33 | mnfxr 7790 | . . . . . 6 | |
34 | ltxr 9517 | . . . . . 6 | |
35 | 33, 33, 34 | mp2an 422 | . . . . 5 |
36 | 32, 35 | mtbir 645 | . . . 4 |
37 | breq12 3904 | . . . . 5 | |
38 | 37 | anidms 394 | . . . 4 |
39 | 36, 38 | mtbiri 649 | . . 3 |
40 | 2, 21, 39 | 3jaoi 1266 | . 2 |
41 | 1, 40 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 class class class wbr 3899 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-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-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 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-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 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-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 |
This theorem is referenced by: xrltnsym 9534 xrltso 9537 xrlttri3 9538 xrleid 9541 xrltne 9551 nltpnft 9552 ngtmnft 9555 xrrebnd 9557 xposdif 9620 lbioog 9651 ubioog 9652 xrmaxleim 10968 xrmaxiflemlub 10972 |
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