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Mirrors > Home > MPE Home > Th. List > eqled | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eqled.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
eqled.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eqled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqled.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | eqled.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | eqle 10139 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-pre-lttri 10010 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 |
This theorem is referenced by: dnibndlem6 32473 int-eqineqd 38493 sublevolico 40201 fourierdlem10 40334 fourierdlem12 40336 fourierdlem37 40361 fourierdlem48 40371 fourierdlem54 40377 fourierdlem79 40402 ioorrnopnxrlem 40526 hoidmvval0b 40804 hoidmv1lelem1 40805 hoidmvlelem2 40810 ovnhoi 40817 volico2 40855 ovolval5lem2 40867 vonioolem2 40895 lighneallem2 41523 |
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