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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 11306 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
2 | breq2 5142 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ 𝐵)) | |
3 | 2 | biimpac 478 | . 2 ⊢ ((𝐴 ≤ 𝐴 ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) |
4 | 1, 3 | sylan 579 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 ℝcr 11104 ≤ cle 11245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-pre-lttri 11179 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 |
This theorem is referenced by: eqled 11313 pfxsuffeqwrdeq 14644 sqrtneglem 15209 leabs 15242 dvlip 25847 nmlno0lem 30481 nmblolbii 30487 nmlnop0iALT 31683 nmbdoplbi 31712 nmcoplbi 31716 nmbdfnlbi 31737 nmcfnlbi 31740 pjnmopi 31836 areacirc 37037 dvconstbi 43548 binomcxplemnn0 43563 |
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