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Mirrors > Home > MPE Home > Th. List > eqlei2 | Structured version Visualization version GIF version |
Description: Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
eqlei2 | ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
2 | eleq1a 2908 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → 𝐵 ∈ ℝ)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐵 ∈ ℝ) |
4 | eqcom 2828 | . . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | letri3 10725 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
6 | 1, 5 | mpan 688 | . . . 4 ⊢ (𝐵 ∈ ℝ → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
7 | 4, 6 | syl5bb 285 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵 = 𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
8 | simpr 487 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
9 | 7, 8 | syl6bi 255 | . 2 ⊢ (𝐵 ∈ ℝ → (𝐵 = 𝐴 → 𝐵 ≤ 𝐴)) |
10 | 3, 9 | mpcom 38 | 1 ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ℝcr 10535 ≤ cle 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 |
This theorem is referenced by: usgruspgr 26962 konigsbergssiedgw 28028 fourierswlem 42514 |
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