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Mirrors > Home > MPE Home > Th. List > prsref | Structured version Visualization version GIF version |
Description: "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
isprs.b | ⊢ 𝐵 = (Base‘𝐾) |
isprs.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
prsref | ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
2 | 1, 1, 1 | 3jca 1128 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
3 | isprs.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
4 | isprs.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | prslem 18247 | . . 3 ⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑋) → 𝑋 ≤ 𝑋))) |
6 | 2, 5 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑋) → 𝑋 ≤ 𝑋))) |
7 | 6 | simpld 495 | 1 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 Basecbs 17140 lecple 17200 Proset cproset 18242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-proset 18244 |
This theorem is referenced by: posref 18267 mgccole1 32147 mgccole2 32148 prsdm 32882 prsrn 32883 prsthinc 47627 |
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