![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pltne | Structured version Visualization version GIF version |
Description: The "less than" relation is not reflexive. (df-pss 3964 analog.) (Contributed by NM, 2-Dec-2011.) |
Ref | Expression |
---|---|
pltne.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltne | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≠ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | pltne.s | . . . 4 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltval 18317 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋 ≠ 𝑌))) |
4 | 3 | simplbda 499 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 < 𝑌) → 𝑋 ≠ 𝑌) |
5 | 4 | ex 412 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≠ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 ‘cfv 6542 lecple 17233 ltcplt 18293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-plt 18315 |
This theorem is referenced by: pltirr 18320 ogrpaddlt 32791 ornglmullt 33016 orngrmullt 33017 ofldchr 33023 isarchiofld 33026 atlen0 38776 1cvratex 38940 ps-2 38945 lhpn0 39471 |
Copyright terms: Public domain | W3C validator |