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Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3983 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
pltval.l | ⊢ ≤ = (le‘𝐾) |
pltval.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltval | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
2 | pltval.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltfval 18389 | . . . 4 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
4 | 3 | breqd 5159 | . . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋 < 𝑌 ↔ 𝑋( ≤ ∖ I )𝑌)) |
5 | brdif 5201 | . . . 4 ⊢ (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
6 | ideqg 5865 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐶 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
7 | 6 | necon3bbid 2976 | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
9 | 8 | anbi2d 630 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 5, 9 | bitrid 283 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 4, 10 | sylan9bb 509 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | 11 | 3impb 1114 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 class class class wbr 5148 I cid 5582 ‘cfv 6563 lecple 17305 ltcplt 18366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-plt 18388 |
This theorem is referenced by: pltle 18391 pltne 18392 pleval2i 18394 pltnle 18396 pltval3 18397 plttr 18400 latnlemlt 18530 latnle 18531 ipolt 18593 ogrpaddlt 33077 ogrpsublt 33081 ornglmullt 33317 orngrmullt 33318 orngmullt 33319 ofldlt1 33323 opltn0 39172 cvrval2 39256 cvrnbtwn2 39257 cvrnbtwn3 39258 cvrle 39260 cvrnbtwn4 39261 cvrne 39263 atlltn0 39288 hlrelat5N 39384 llnle 39501 lplnle 39523 llncvrlpln2 39540 lplncvrlvol2 39598 lhp2lt 39984 lautlt 40074 |
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