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Mirrors > Home > ILE Home > Th. List > elxr | Unicode version |
Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
elxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xr 7772 | . . 3 | |
2 | 1 | eleq2i 2184 | . 2 |
3 | elun 3187 | . 2 | |
4 | pnfex 7787 | . . . . 5 | |
5 | mnfxr 7790 | . . . . . 6 | |
6 | 5 | elexi 2672 | . . . . 5 |
7 | 4, 6 | elpr2 3519 | . . . 4 |
8 | 7 | orbi2i 736 | . . 3 |
9 | 3orass 950 | . . 3 | |
10 | 8, 9 | bitr4i 186 | . 2 |
11 | 2, 3, 10 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 cun 3039 cpr 3498 cr 7587 cpnf 7765 cmnf 7766 cxr 7767 |
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-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-un 4325 ax-cnex 7679 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 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-uni 3707 df-pnf 7770 df-mnf 7771 df-xr 7772 |
This theorem is referenced by: xrnemnf 9519 xrnepnf 9520 xrltnr 9521 xrltnsym 9534 xrlttr 9536 xrltso 9537 xrlttri3 9538 nltpnft 9552 npnflt 9553 ngtmnft 9555 nmnfgt 9556 xrrebnd 9557 xnegcl 9570 xnegneg 9571 xltnegi 9573 xrpnfdc 9580 xrmnfdc 9581 xnegid 9597 xaddcom 9599 xaddid1 9600 xnegdi 9606 xleadd1a 9611 xltadd1 9614 xlt2add 9618 xsubge0 9619 xposdif 9620 xleaddadd 9625 qbtwnxr 9990 xrmaxiflemcl 10969 xrmaxifle 10970 xrmaxiflemab 10971 xrmaxiflemlub 10972 xrmaxltsup 10982 xrmaxadd 10985 xrbdtri 11000 isxmet2d 12428 blssioo 12625 |
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