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Mirrors > Home > MPE Home > Th. List > pltirr | Structured version Visualization version GIF version |
Description: The "less than" relation is not reflexive. (pssirr 4096 analog.) (Contributed by NM, 7-Feb-2012.) |
Ref | Expression |
---|---|
pltne.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltirr | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋 < 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ 𝑋 = 𝑋 | |
2 | pltne.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
3 | 2 | pltne 18269 | . . . 4 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 < 𝑋 → 𝑋 ≠ 𝑋)) |
4 | 3 | 3anidm23 1421 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 < 𝑋 → 𝑋 ≠ 𝑋)) |
5 | 4 | necon2bd 2955 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑋 → ¬ 𝑋 < 𝑋)) |
6 | 1, 5 | mpi 20 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋 < 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5141 ‘cfv 6532 ltcplt 18243 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fv 6540 df-plt 18265 |
This theorem is referenced by: pospo 18280 atnlt 37986 llnnlt 38197 lplnnlt 38239 lvolnltN 38292 |
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