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Mirrors > Home > MPE Home > Th. List > leidi | Structured version Visualization version GIF version |
Description: 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
lt2.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
leidi | ⊢ 𝐴 ≤ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | leid 11071 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 class class class wbr 5074 ℝcr 10870 ≤ cle 11010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-pre-lttri 10945 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 |
This theorem is referenced by: 1le1 11603 elimge0 11814 lemul1a 11829 0le0 12074 dfuzi 12411 fldiv4p1lem1div2 13555 facwordi 14003 sincos2sgn 15903 strle1 16859 cnfldfunALTOLD 20611 dscmet 23728 tanabsge 25663 logneg 25743 log2ublem2 26097 emcllem6 26150 harmonicbnd3 26157 ppiublem2 26351 chebbnd1lem3 26619 rpvmasumlem 26635 axlowdimlem6 27315 umgrupgr 27473 umgrislfupgr 27493 usgrislfuspgr 27554 usgr2pthlem 28131 konigsberglem4 28619 pfx1s2 31213 lmat22e12 31769 lmat22e21 31770 lmat22e22 31771 oddpwdc 32321 tgoldbachgt 32643 bj-pinftynminfty 35398 lhe4.4ex1a 41947 limsup10exlem 43313 fourierdlem112 43759 salexct3 43881 salgensscntex 43883 0ome 44067 wtgoldbnnsum4prm 45254 bgoldbnnsum3prm 45256 |
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