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Mirrors > Home > MPE Home > Th. List > eqle | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
Ref | Expression |
---|---|
eqle | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leid 10738 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
2 | breq2 5072 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ 𝐵)) | |
3 | 2 | biimpac 481 | . 2 ⊢ ((𝐴 ≤ 𝐴 ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) |
4 | 1, 3 | sylan 582 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝcr 10538 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-pre-lttri 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: eqled 10745 pfxsuffeqwrdeq 14062 sqrtneglem 14628 leabs 14661 dvlip 24592 nmlno0lem 28572 nmblolbii 28578 nmlnop0iALT 29774 nmbdoplbi 29803 nmcoplbi 29807 nmbdfnlbi 29828 nmcfnlbi 29831 pjnmopi 29927 areacirc 34989 dvconstbi 40673 binomcxplemnn0 40688 |
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