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| Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version | ||
| Description: Less-than relation. (df-pss 3971 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 18376 | . . . 4 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
| 4 | 3 | breqd 5154 | . . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋 < 𝑌 ↔ 𝑋( ≤ ∖ I )𝑌)) |
| 5 | brdif 5196 | . . . 4 ⊢ (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
| 6 | ideqg 5862 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐶 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
| 7 | 6 | necon3bbid 2978 | . . . . . 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 1115 | 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 class class class wbr 5143 I cid 5577 ‘cfv 6561 lecple 17304 ltcplt 18354 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-plt 18375 |
| This theorem is referenced by: pltle 18378 pltne 18379 pleval2i 18381 pltnle 18383 pltval3 18384 plttr 18387 latnlemlt 18517 latnle 18518 ipolt 18580 ogrpaddlt 33094 ogrpsublt 33098 ornglmullt 33337 orngrmullt 33338 orngmullt 33339 ofldlt1 33343 opltn0 39191 cvrval2 39275 cvrnbtwn2 39276 cvrnbtwn3 39277 cvrle 39279 cvrnbtwn4 39280 cvrne 39282 atlltn0 39307 hlrelat5N 39403 llnle 39520 lplnle 39542 llncvrlpln2 39559 lplncvrlvol2 39617 lhp2lt 40003 lautlt 40093 |
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