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Mirrors > Home > MPE Home > Th. List > ltnr | Structured version Visualization version GIF version |
Description: 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
ltnr | ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 11333 | . 2 ⊢ < Or ℝ | |
2 | sonr 5608 | . 2 ⊢ (( < Or ℝ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 < 𝐴) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 class class class wbr 5144 Or wor 5584 ℝcr 11146 < clt 11287 |
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 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-pre-lttri 11221 ax-pre-lttrn 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-ltxr 11292 |
This theorem is referenced by: ltne 11350 ltnri 11362 ltnrd 11387 squeeze0 12161 xrltnr 13145 |
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