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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 18342 | . 2 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝐵)) |
8 | ple1.1 | . . . 4 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 3, 8 | p1val 18411 | . . 3 ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵)) |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → 1 = (𝑈‘𝐵)) |
11 | 7, 10 | breqtrrd 5170 | 1 ⊢ (𝜑 → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 dom cdm 5672 ‘cfv 6542 Basecbs 17171 lecple 17231 lubclub 18292 1.cp1 18407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-lub 18329 df-p1 18409 |
This theorem is referenced by: ople1 38600 lhp2lt 39411 |
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