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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 44815 | . . 3 ⊢ > = ◡ < | |
2 | 1 | breqi 5065 | . 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵) |
3 | gt-lth.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gt-lth.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5748 | . 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴) |
6 | 2, 5 | bitri 277 | 1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 ◡ccnv 5549 < clt 10669 > cgt 44813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-cnv 5558 df-gt 44815 |
This theorem is referenced by: ex-gt 44820 |
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