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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 8003 | . 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 2141 class class class wbr 3989 ℝcr 7773 ≤ cle 7955 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 |
This theorem is referenced by: zextle 9303 uzind 9323 uzid 9501 z2ge 9783 nn0fz0 10075 fvinim0ffz 10197 flid 10240 modqabs2 10314 monoord 10432 leexp2r 10530 facwordi 10674 faclbnd6 10678 sqrtgt0 10998 abs00ap 11026 isumlessdc 11459 cvgratnnlemnexp 11487 cvgratnnlemmn 11488 eirraplem 11739 nn0seqcvgd 11995 pcidlem 12276 pc2dvds 12283 pcprmpw2 12286 pcmpt 12295 trilpolemclim 14068 trilpolemisumle 14070 trilpolemeq1 14072 |
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