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Mirrors > Home > MPE Home > Th. List > posasymb | Structured version Visualization version GIF version |
Description: A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
posi.b | ⊢ 𝐵 = (Base‘𝐾) |
posi.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
posasymb | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Poset) | |
2 | simp2 1055 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | simp3 1056 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
4 | posi.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | posi.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
6 | 4, 5 | posi 16773 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌) → 𝑋 ≤ 𝑌))) |
7 | 1, 2, 3, 3, 6 | syl13anc 1320 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌) → 𝑋 ≤ 𝑌))) |
8 | 7 | simp2d 1067 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌)) |
9 | 4, 5 | posref 16774 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
10 | breq2 4587 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌)) | |
11 | 9, 10 | syl5ibcom 234 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌)) |
12 | breq1 4586 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋)) | |
13 | 9, 12 | syl5ibcom 234 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋)) |
14 | 11, 13 | jcad 554 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑌 → (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋))) |
15 | 14 | 3adant3 1074 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋))) |
16 | 8, 15 | impbid 201 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-preset 16751 df-poset 16769 |
This theorem is referenced by: pltnle 16789 pltval3 16790 lublecllem 16811 latasymb 16877 latleeqj1 16886 latleeqm1 16902 odupos 16958 poslubmo 16969 posglbmo 16970 posrasymb 28988 archirngz 29074 archiabllem1a 29076 ople0 33492 op1le 33497 atlle0 33610 |
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