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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 2820 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → 𝐵 ∈ ℝ)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐵 ∈ ℝ) |
4 | eqcom 2732 | . . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | letri3 11331 | . . . . 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 | bitrid 282 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵 = 𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
8 | simpr 483 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴) | |
9 | 7, 8 | biimtrdi 252 | . 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ℝcr 11139 ≤ cle 11281 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 |
This theorem is referenced by: usgruspgr 29065 konigsbergssiedgw 30132 fourierswlem 45756 |
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