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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 48231 | . . 3 ⊢ ≥ = ◡ ≤ | |
2 | 1 | breqi 5158 | . 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵) |
3 | gte-lteh.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gte-lteh.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5889 | . 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴) |
6 | 2, 5 | bitri 274 | 1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 ◡ccnv 5681 ≤ cle 11287 ≥ cge-real 48229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-cnv 5690 df-gte 48231 |
This theorem is referenced by: ex-gte 48238 |
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