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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 8030 | . 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 2148 class class class wbr 4002 ℝcr 7807 < clt 7988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-pre-ltirr 7920 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-pnf 7990 df-mnf 7991 df-ltxr 7993 |
This theorem is referenced by: fzonel 10155 frec2uzlt2d 10399 frec2uzf1od 10401 zfz1isolemiso 10812 recvguniqlem 10996 resqrexlemoverl 11023 leabs 11076 ltabs 11089 maxleim 11207 climuni 11294 infssuzex 11942 znnen 12391 dedekindeulemeu 13971 dedekindicclemeu 13980 ivthinc 13992 limcimo 14005 efltlemlt 14066 taupi 14680 |
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