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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 17597 | . 2 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝐵)) |
8 | ple1.1 | . . . 4 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 3, 8 | p1val 17652 | . . 3 ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵)) |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → 1 = (𝑈‘𝐵)) |
11 | 7, 10 | breqtrrd 5094 | 1 ⊢ (𝜑 → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 Basecbs 16483 lecple 16572 lubclub 17552 1.cp1 17648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-lub 17584 df-p1 17650 |
This theorem is referenced by: ople1 36342 lhp2lt 37152 |
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