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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 7773 | . 2 | |
2 | df-3an 949 | . . . . . 6 | |
3 | 2 | opabbii 3965 | . . . . 5 |
4 | opabssxp 4583 | . . . . 5 | |
5 | 3, 4 | eqsstri 3099 | . . . 4 |
6 | rexpssxrxp 7778 | . . . 4 | |
7 | 5, 6 | sstri 3076 | . . 3 |
8 | ressxr 7777 | . . . . . 6 | |
9 | snsspr2 3639 | . . . . . . 7 | |
10 | ssun2 3210 | . . . . . . . 8 | |
11 | df-xr 7772 | . . . . . . . 8 | |
12 | 10, 11 | sseqtrri 3102 | . . . . . . 7 |
13 | 9, 12 | sstri 3076 | . . . . . 6 |
14 | 8, 13 | unssi 3221 | . . . . 5 |
15 | snsspr1 3638 | . . . . . 6 | |
16 | 15, 12 | sstri 3076 | . . . . 5 |
17 | xpss12 4616 | . . . . 5 | |
18 | 14, 16, 17 | mp2an 422 | . . . 4 |
19 | xpss12 4616 | . . . . 5 | |
20 | 13, 8, 19 | mp2an 422 | . . . 4 |
21 | 18, 20 | unssi 3221 | . . 3 |
22 | 7, 21 | unssi 3221 | . 2 |
23 | 1, 22 | eqsstri 3099 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 w3a 947 wcel 1465 cun 3039 wss 3041 csn 3497 cpr 3498 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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pr 3504 df-opab 3960 df-xp 4515 df-xr 7772 df-ltxr 7773 |
This theorem is referenced by: ltrel 7794 |
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