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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leeq1d | Structured version Visualization version GIF version |
Description: Specialization of breq1d 5119 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
leeq1d.1 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
leeq1d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
leeq1d.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leeq1d.4 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
leeq1d | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leeq1d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | leeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
3 | 1, 2 | eqbrtrrd 5133 | 1 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5109 ℝcr 11058 ≤ cle 11198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 |
This theorem is referenced by: imo72b2lem0 42530 |
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