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Mirrors > Home > MPE Home > Th. List > ple1 | Structured version Visualization version GIF version |
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
ple1.b | ⊢ 𝐵 = (Base‘𝐾) |
ple1.u | ⊢ 𝑈 = (lub‘𝐾) |
ple1.l | ⊢ ≤ = (le‘𝐾) |
ple1.1 | ⊢ 1 = (1.‘𝐾) |
ple1.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
ple1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ple1.d | ⊢ (𝜑 → 𝐵 ∈ dom 𝑈) |
Ref | Expression |
---|---|
ple1 | ⊢ (𝜑 → 𝑋 ≤ 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ple1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ple1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | ple1.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
4 | ple1.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | ple1.d | . . 3 ⊢ (𝜑 → 𝐵 ∈ dom 𝑈) | |
6 | ple1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | luble 17865 | . 2 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝐵)) |
8 | ple1.1 | . . . 4 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 3, 8 | p1val 17934 | . . 3 ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵)) |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → 1 = (𝑈‘𝐵)) |
11 | 7, 10 | breqtrrd 5081 | 1 ⊢ (𝜑 → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 dom cdm 5551 ‘cfv 6380 Basecbs 16760 lecple 16809 lubclub 17816 1.cp1 17930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-lub 17852 df-p1 17932 |
This theorem is referenced by: ople1 36942 lhp2lt 37752 |
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