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Mirrors > Home > MPE Home > Th. List > Mathboxes > leeq2d | Structured version Visualization version GIF version |
Description: Specialization of breq2d 4665 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 4679 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 |
This theorem is referenced by: imo72b2lem0 38465 |
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