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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xr 7219 |
. . 3
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2 | 1 | eleq2i 2146 |
. 2
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3 | elun 3114 |
. 2
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4 | pnfex 7234 |
. . . . 5
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5 | mnfxr 7237 |
. . . . . 6
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6 | 5 | elexi 2612 |
. . . . 5
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7 | 4, 6 | elpr2 3428 |
. . . 4
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8 | 7 | orbi2i 712 |
. . 3
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9 | 3orass 923 |
. . 3
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10 | 8, 9 | bitr4i 185 |
. 2
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11 | 2, 3, 10 | 3bitri 204 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-un 4196 ax-cnex 7129 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-rex 2355 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-uni 3610 df-pnf 7217 df-mnf 7218 df-xr 7219 |
This theorem is referenced by: xrnemnf 8929 xrnepnf 8930 xrltnr 8931 xrltnsym 8944 xrlttr 8946 xrltso 8947 xrlttri3 8948 nltpnft 8960 ngtmnft 8961 xrrebnd 8962 xnegcl 8975 xnegneg 8976 xltnegi 8978 qbtwnxr 9344 |
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