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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 10459 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 class class class wbr 4875 ℝcr 10258 ≤ cle 10399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-pre-lttri 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 |
This theorem is referenced by: 1le1 10987 elimge0 11197 lemul1a 11214 0le0 11466 dfuzi 11803 fldiv4p1lem1div2 12938 facwordi 13376 sincos2sgn 15303 strle1 16339 cnfldfun 20125 dscmet 22754 tanabsge 24665 logneg 24740 log2ublem2 25094 emcllem6 25147 harmonicbnd3 25154 ppiublem2 25348 chebbnd1lem3 25580 rpvmasumlem 25596 axlowdimlem6 26253 umgrupgr 26408 umgrislfupgr 26428 usgrislfuspgr 26490 usgr2pthlem 27072 konigsberglem4 27630 lmat22e12 30426 lmat22e21 30427 lmat22e22 30428 oddpwdc 30957 tgoldbachgt 31286 bj-pinftynminfty 33649 lhe4.4ex1a 39363 limsup10exlem 40793 fourierdlem112 41223 salexct3 41345 salgensscntex 41347 0ome 41531 wtgoldbnnsum4prm 42534 bgoldbnnsum3prm 42536 |
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