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Mirrors > Home > ILE Home > Th. List > xrre3 | Unicode version |
Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
xrre3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt 9813 |
. . . . . 6
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2 | 1 | adantl 277 |
. . . . 5
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3 | mnfxr 8044 |
. . . . . . 7
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4 | 3 | a1i 9 |
. . . . . 6
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5 | rexr 8033 |
. . . . . . 7
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6 | 5 | adantl 277 |
. . . . . 6
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7 | simpl 109 |
. . . . . 6
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8 | xrltletr 9837 |
. . . . . 6
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9 | 4, 6, 7, 8 | syl3anc 1249 |
. . . . 5
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10 | 2, 9 | mpand 429 |
. . . 4
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11 | 10 | imp 124 |
. . 3
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12 | 11 | adantrr 479 |
. 2
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13 | simprr 531 |
. 2
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14 | xrrebnd 9849 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | ad2antrr 488 |
. 2
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16 | 12, 13, 15 | mpbir2and 946 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-po 4314 df-iso 4315 df-xp 4650 df-cnv 4652 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 |
This theorem is referenced by: elicore 10297 |
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