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Mirrors > Home > ILE Home > Th. List > lensymd | GIF version |
Description: 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lensymd.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
lensymd | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lensymd.3 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | lenltd 8062 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbid 147 | 1 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2148 class class class wbr 4000 ℝcr 7798 < clt 7979 ≤ cle 7980 |
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-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-xr 7983 df-le 7985 |
This theorem is referenced by: lbinf 8891 addmodlteq 10381 iseqf1olemab 10472 seq3f1olemqsumk 10482 seq3f1olemqsum 10483 nn0ltexp2 10671 zfz1isolemiso 10800 seq3coll 10803 maxleim 11195 maxabslemval 11198 cvgratz 11521 divalglemnqt 11905 suprzubdc 11933 zsupssdc 11935 bezoutlemsup 11990 dfgcd2 11995 lcmgcdlem 12057 lgsval2lem 14071 trilpolemgt1 14436 trilpolemlt1 14438 |
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