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Mirrors > Home > MPE Home > Th. List > Mathboxes > gt-lth | Structured version Visualization version GIF version |
Description: Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
gt-lth.1 | ⊢ 𝐴 ∈ V |
gt-lth.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
gt-lth | ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gt 48154 | . . 3 ⊢ > = ◡ < | |
2 | 1 | breqi 5154 | . 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵) |
3 | gt-lth.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gt-lth.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5885 | . 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 2099 Vcvv 3471 class class class wbr 5148 ◡ccnv 5677 < clt 11279 > cgt 48152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-cnv 5686 df-gt 48154 |
This theorem is referenced by: ex-gt 48159 |
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