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Mirrors > Home > MPE Home > Th. List > pleval2i | Structured version Visualization version GIF version |
Description: One direction of pleval2 18218. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
pleval2.b | ⊢ 𝐵 = (Base‘𝐾) |
pleval2.l | ⊢ ≤ = (le‘𝐾) |
pleval2.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pleval2i | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6876 | . . . . . . . . 9 ⊢ (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base) | |
2 | pleval2.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾) | |
3 | 1, 2 | eleq2s 2856 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝐾 ∈ dom Base) |
4 | 3 | adantr 481 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ dom Base) |
5 | pleval2.l | . . . . . . . . 9 ⊢ ≤ = (le‘𝐾) | |
6 | pleval2.s | . . . . . . . . 9 ⊢ < = (lt‘𝐾) | |
7 | 5, 6 | pltval 18213 | . . . . . . . 8 ⊢ ((𝐾 ∈ dom Base ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
8 | 7 | 3expb 1120 | . . . . . . 7 ⊢ ((𝐾 ∈ dom Base ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
9 | 4, 8 | mpancom 686 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 9 | biimpar 478 | . . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑋 < 𝑌) |
11 | 10 | expr 457 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 → 𝑋 < 𝑌)) |
12 | 11 | necon1bd 2959 | . . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (¬ 𝑋 < 𝑌 → 𝑋 = 𝑌)) |
13 | 12 | orrd 861 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)) |
14 | 13 | ex 413 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5103 dom cdm 5631 ‘cfv 6493 Basecbs 17075 lecple 17132 ltcplt 18189 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-plt 18211 |
This theorem is referenced by: pleval2 18218 pospo 18226 |
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