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Mirrors > Home > MPE Home > Th. List > pleval2 | Structured version Visualization version GIF version |
Description: Less-than-or-equal in terms of less-than. (sspss 3668 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
pleval2.b | ⊢ 𝐵 = (Base‘𝐾) |
pleval2.l | ⊢ ≤ = (le‘𝐾) |
pleval2.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pleval2 | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pleval2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pleval2.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | pleval2.s | . . . 4 ⊢ < = (lt‘𝐾) | |
4 | 1, 2, 3 | pleval2i 16787 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
5 | 4 | 3adant1 1072 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
6 | 2, 3 | pltle 16784 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
7 | 1, 2 | posref 16774 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
8 | 7 | 3adant3 1074 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
9 | breq2 4587 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌)) | |
10 | 8, 9 | syl5ibcom 234 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌)) |
11 | 6, 10 | jaod 394 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 < 𝑌 ∨ 𝑋 = 𝑌) → 𝑋 ≤ 𝑌)) |
12 | 5, 11 | impbid 201 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 ltcplt 16764 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 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-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-preset 16751 df-poset 16769 df-plt 16781 |
This theorem is referenced by: pltletr 16794 plelttr 16795 tosso 16859 tlt3 28996 orngsqr 29135 |
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