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Mirrors > Home > MPE Home > Th. List > postr | Structured version Visualization version GIF version |
Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.) |
Ref | Expression |
---|---|
posi.b | ⊢ 𝐵 = (Base‘𝐾) |
posi.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
postr | ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | posi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | posi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | posi 17560 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) |
4 | 3 | simp3d 1140 | 1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Posetcpo 17550 |
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-nul 5210 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-poset 17556 |
This theorem is referenced by: plttr 17580 joinle 17624 meetle 17638 lattr 17666 odupos 17745 omndadd2d 30709 omndadd2rd 30710 omndmul2 30713 atlatle 36471 cvratlem 36572 llncmp 36673 llncvrlpln 36709 lplncmp 36713 lplncvrlvol 36767 lvolcmp 36768 pmaple 36912 |
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