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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-ineqtransd | Structured version Visualization version GIF version |
Description: InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-ineqtransd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
int-ineqtransd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-ineqtransd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-ineqtransd.4 | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
int-ineqtransd.5 | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
int-ineqtransd | ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-ineqtransd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | int-ineqtransd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | int-ineqtransd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | int-ineqtransd.5 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
5 | int-ineqtransd.4 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
6 | 1, 2, 3, 4, 5 | letrd 10786 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 ≤ cle 10665 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: (None) |
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