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Mirrors > Home > ILE Home > Th. List > leidd | GIF version |
Description: 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
leidd | ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leid 7962 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 class class class wbr 3966 ℝcr 7732 ≤ cle 7914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltirr 7845 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-cnv 4595 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 |
This theorem is referenced by: zextle 9256 uzind 9276 uzid 9454 z2ge 9731 nn0fz0 10022 fvinim0ffz 10144 flid 10187 modqabs2 10261 monoord 10379 leexp2r 10477 facwordi 10618 faclbnd6 10622 sqrtgt0 10938 abs00ap 10966 isumlessdc 11397 cvgratnnlemnexp 11425 cvgratnnlemmn 11426 eirraplem 11677 nn0seqcvgd 11922 trilpolemclim 13649 trilpolemisumle 13651 trilpolemeq1 13653 |
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