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 5067 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 5081 | 1 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 |
This theorem is referenced by: imo72b2lem0 40394 |
Copyright terms: Public domain | W3C validator |