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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leeq2d | Structured version Visualization version GIF version |
Description: Specialization of breq2d 5117 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 5131 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ℝcr 11049 ≤ cle 11189 |
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 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 |
This theorem is referenced by: imo72b2lem0 42419 |
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