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Mathbox for David A. Wheeler |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gte-lteh | Structured version Visualization version GIF version |
Description: Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
gte-lteh.1 | ⊢ 𝐴 ∈ V |
gte-lteh.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
gte-lteh | ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gte 44222 | . . 3 ⊢ ≥ = ◡ ≤ | |
2 | 1 | breqi 4931 | . 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵) |
3 | gte-lteh.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gte-lteh.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5599 | . 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴) |
6 | 2, 5 | bitri 267 | 1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2051 Vcvv 3408 class class class wbr 4925 ◡ccnv 5402 ≤ cle 10473 ≥ cge-real 44220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-cnv 5411 df-gte 44222 |
This theorem is referenced by: ex-gte 44229 |
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