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Mirrors > Home > MPE Home > Th. List > ltnsymd | Structured version Visualization version GIF version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltled.1 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltnsymd | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltled.1 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐵) | |
4 | 1, 2, 3 | ltled 11299 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
5 | 1, 2 | lenltd 11297 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
6 | 4, 5 | mpbid 231 | 1 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 class class class wbr 5103 ℝcr 11046 < clt 11185 ≤ cle 11186 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-pre-lttri 11121 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 |
This theorem is referenced by: fvmptnn04ifd 22186 chfacfscmulgsum 22193 chfacfpmmulgsum 22197 bposlem9 26624 ostth2lem1 26950 tgcgr4 27359 signsvtp 33064 dffltz 40910 rpnnen3lem 41293 limcrecl 43802 icccncfext 44060 fourierdlem10 44290 fourierdlem40 44320 fourierdlem74 44353 fourierdlem75 44354 fourierdlem78 44357 fourierdlem103 44382 sqwvfoura 44401 sqwvfourb 44402 fourierswlem 44403 |
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