![]() |
Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > leeq1d | Structured version Visualization version GIF version |
Description: Specialization of breq1d 5113 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 5127 | 1 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ℝcr 11046 ≤ 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-ext 2707 |
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-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 |
This theorem is referenced by: imo72b2lem0 42380 |
Copyright terms: Public domain | W3C validator |