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Mirrors > Home > MPE Home > Th. List > Mathboxes > leeq2d | Structured version Visualization version GIF version |
Description: Specialization of breq2d 5077 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
leeq2d.1 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
leeq2d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
leeq2d.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leeq2d.4 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
leeq2d | ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leeq2d.1 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
2 | leeq2d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | 1, 2 | breqtrd 5091 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ℝcr 10535 ≤ cle 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 |
This theorem is referenced by: imo72b2lem0 40514 |
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