Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xreqled | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xreqled.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xreqled.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
xreqled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xreqled.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xreqled.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | xreqle 42387 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 class class class wbr 5031 ℝ*cxr 10753 ≤ cle 10755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-pre-lttri 10690 ax-pre-lttrn 10691 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 |
This theorem is referenced by: suplesup 42408 infleinf 42441 infxrpnf 42517 liminfgelimsup 42857 liminfgelimsupuz 42863 climliminflimsup 42883 hoicvrrex 43628 ovolval5lem1 43724 |
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