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Mirrors > Home > ILE Home > Th. List > ltrelxr | Unicode version |
Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ltrelxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltxr 7065 | . 2 | |
2 | df-3an 887 | . . . . . 6 | |
3 | 2 | opabbii 3824 | . . . . 5 |
4 | opabssxp 4414 | . . . . 5 | |
5 | 3, 4 | eqsstri 2975 | . . . 4 |
6 | rexpssxrxp 7070 | . . . 4 | |
7 | 5, 6 | sstri 2954 | . . 3 |
8 | ressxr 7069 | . . . . . 6 | |
9 | snsspr2 3513 | . . . . . . 7 | |
10 | ssun2 3107 | . . . . . . . 8 | |
11 | df-xr 7064 | . . . . . . . 8 | |
12 | 10, 11 | sseqtr4i 2978 | . . . . . . 7 |
13 | 9, 12 | sstri 2954 | . . . . . 6 |
14 | 8, 13 | unssi 3118 | . . . . 5 |
15 | snsspr1 3512 | . . . . . 6 | |
16 | 15, 12 | sstri 2954 | . . . . 5 |
17 | xpss12 4445 | . . . . 5 | |
18 | 14, 16, 17 | mp2an 402 | . . . 4 |
19 | xpss12 4445 | . . . . 5 | |
20 | 13, 8, 19 | mp2an 402 | . . . 4 |
21 | 18, 20 | unssi 3118 | . . 3 |
22 | 7, 21 | unssi 3118 | . 2 |
23 | 1, 22 | eqsstri 2975 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 w3a 885 wcel 1393 cun 2915 wss 2917 csn 3375 cpr 3376 class class class wbr 3764 copab 3817 cxp 4343 cr 6888 cltrr 6893 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pr 3382 df-opab 3819 df-xp 4351 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: ltrel 7081 |
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