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Mirrors > Home > MPE Home > Th. List > letsr | Unicode version |
Description: The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
letsr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lerel 9106 |
. . 3
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2 | lerelxr 9105 |
. . . . . . . . . . 11
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3 | 2 | brel 4893 |
. . . . . . . . . 10
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4 | 3 | adantr 452 |
. . . . . . . . 9
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5 | 4 | simpld 446 |
. . . . . . . 8
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6 | 4 | simprd 450 |
. . . . . . . 8
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7 | 2 | brel 4893 |
. . . . . . . . . 10
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8 | 7 | simprd 450 |
. . . . . . . . 9
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9 | 8 | adantl 453 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 5, 6, 9 | 3jca 1134 |
. . . . . . 7
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11 | xrletr 10712 |
. . . . . . 7
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12 | 10, 11 | mpcom 34 |
. . . . . 6
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13 | 12 | ax-gen 1552 |
. . . . 5
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14 | 13 | gen2 1553 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | cotr 5213 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | mpbir 201 |
. . 3
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17 | asymref 5217 |
. . . 4
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18 | simpr 448 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 2 | brel 4893 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | simpld 446 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | adantl 453 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | xrletri3 10709 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | sylan2 461 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | mpbird 224 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | ex 424 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | xrleid 10707 |
. . . . . . . . 9
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27 | 26, 26 | jca 519 |
. . . . . . . 8
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28 | breq2 4184 |
. . . . . . . . 9
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29 | breq1 4183 |
. . . . . . . . 9
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30 | 28, 29 | anbi12d 692 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 27, 30 | syl5ibcom 212 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 25, 31 | impbid 184 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32 | alrimiv 1638 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | lefld 14634 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 34 | eqcomi 2416 |
. . . . 5
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36 | 33, 35 | eleq2s 2504 |
. . . 4
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37 | 17, 36 | mprgbir 2744 |
. . 3
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38 | xrex 10573 |
. . . . . 6
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39 | 38, 38 | xpex 4957 |
. . . . 5
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40 | 39, 2 | ssexi 4316 |
. . . 4
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41 | isps 14597 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | 40, 41 | ax-mp 8 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 1, 16, 37, 42 | mpbir3an 1136 |
. 2
![]() ![]() ![]() ![]() |
44 | xrletri 10708 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | 44 | rgen2a 2740 |
. . 3
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46 | qfto 5222 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | mpbir 201 |
. 2
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48 | ledm 14632 |
. . 3
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49 | 48 | istsr 14612 |
. 2
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50 | 43, 47, 49 | mpbir2an 887 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: cnfldle 16675 letopon 17231 leordtval2 17238 leordtval 17239 iccordt 17240 ordtrestixx 17248 xrge0tsms 18826 icopnfhmeo 18929 iccpnfhmeo 18931 xrhmeo 18932 xrhaus 24089 xrge0tsmsd 24184 cnvordtrestixx 24272 xrmulc1cn 24277 xrge0iifhmeo 24283 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-cnex 9010 ax-resscn 9011 ax-pre-lttri 9028 ax-pre-lttrn 9029 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-op 3791 df-uni 3984 df-br 4181 df-opab 4235 df-mpt 4236 df-id 4466 df-po 4471 df-so 4472 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-er 6872 df-en 7077 df-dom 7078 df-sdom 7079 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-ps 14592 df-tsr 14593 |
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