Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3956 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 17571 | . . . 4 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
4 | 3 | breqd 5079 | . . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋 < 𝑌 ↔ 𝑋( ≤ ∖ I )𝑌)) |
5 | brdif 5121 | . . . 4 ⊢ (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
6 | ideqg 5724 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐶 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
7 | 6 | necon3bbid 3055 | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
8 | 7 | adantl 484 | . . . . 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 | syl5bb 285 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 4, 10 | sylan9bb 512 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | 11 | 3impb 1111 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 class class class wbr 5068 I cid 5461 ‘cfv 6357 lecple 16574 ltcplt 17553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-plt 17570 |
This theorem is referenced by: pltle 17573 pltne 17574 pleval2i 17576 pltnle 17578 pltval3 17579 plttr 17582 latnlemlt 17696 latnle 17697 ipolt 17771 ogrpaddlt 30720 ogrpsublt 30724 ornglmullt 30882 orngrmullt 30883 orngmullt 30884 ofldlt1 30888 opltn0 36328 cvrval2 36412 cvrnbtwn2 36413 cvrnbtwn3 36414 cvrle 36416 cvrnbtwn4 36417 cvrne 36419 atlltn0 36444 hlrelat5N 36539 llnle 36656 lplnle 36678 llncvrlpln2 36695 lplncvrlvol2 36753 lhp2lt 37139 lautlt 37229 |
Copyright terms: Public domain | W3C validator |