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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 11348 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 class class class wbr 5152 ℝcr 11145 ≤ cle 11287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-pre-lttri 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 |
This theorem is referenced by: 1le1 11880 elimge0 12091 lemul1a 12106 0le0 12351 dfuzi 12691 fldiv4p1lem1div2 13840 facwordi 14288 sincos2sgn 16178 strle1 17134 cnfldfunALTOLDOLD 21315 dscmet 24501 tanabsge 26461 logneg 26542 log2ublem2 26899 emcllem6 26953 harmonicbnd3 26960 ppiublem2 27156 chebbnd1lem3 27424 rpvmasumlem 27440 axlowdimlem6 28778 umgrupgr 28936 umgrislfupgr 28956 usgrislfuspgr 29020 usgr2pthlem 29597 konigsberglem4 30085 pfx1s2 32683 lmat22e12 33453 lmat22e21 33454 lmat22e22 33455 oddpwdc 34007 tgoldbachgt 34328 bj-pinftynminfty 36739 lhe4.4ex1a 43797 limsup10exlem 45189 fourierdlem112 45635 salexct3 45759 salgensscntex 45761 0ome 45946 wtgoldbnnsum4prm 47171 bgoldbnnsum3prm 47173 |
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