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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 47253 | . . 3 ⊢ ≥ = ◡ ≤ | |
2 | 1 | breqi 5112 | . 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵) |
3 | gte-lteh.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gte-lteh.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5839 | . 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴) |
6 | 2, 5 | bitri 275 | 1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 Vcvv 3444 class class class wbr 5106 ◡ccnv 5633 ≤ cle 11195 ≥ cge-real 47251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-cnv 5642 df-gte 47253 |
This theorem is referenced by: ex-gte 47260 |
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