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Mirrors > Home > MPE Home > Th. List > eqlei | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
eqlei | ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
2 | eleq1a 2885 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → 𝐵 ∈ ℝ)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (𝐵 = 𝐴 → 𝐵 ∈ ℝ) |
4 | 3 | eqcoms 2806 | . 2 ⊢ (𝐴 = 𝐵 → 𝐵 ∈ ℝ) |
5 | letri3 10715 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
6 | 1, 5 | mpan 689 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
7 | simpl 486 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 ≤ 𝐵) | |
8 | 6, 7 | syl6bi 256 | . 2 ⊢ (𝐵 ∈ ℝ → (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)) |
9 | 4, 8 | mpcom 38 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: le2tri3i 10759 fldiv4lem1div2 13202 vdegp1bi 27327 |
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