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Mirrors > Home > ILE Home > Th. List > ltnrd | GIF version |
Description: 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
ltnrd | ⊢ (𝜑 → ¬ 𝐴 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnr 7966 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ¬ 𝐴 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2135 class class class wbr 3976 ℝcr 7743 < clt 7924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-pnf 7926 df-mnf 7927 df-ltxr 7929 |
This theorem is referenced by: fzonel 10085 frec2uzlt2d 10329 frec2uzf1od 10331 zfz1isolemiso 10738 recvguniqlem 10922 resqrexlemoverl 10949 leabs 11002 ltabs 11015 maxleim 11133 climuni 11220 infssuzex 11867 znnen 12274 dedekindeulemeu 13147 dedekindicclemeu 13156 ivthinc 13168 limcimo 13181 efltlemlt 13242 taupi 13790 |
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