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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 44841 | . . 3 ⊢ ≥ = ◡ ≤ | |
2 | 1 | breqi 5072 | . 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵) |
3 | gte-lteh.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | gte-lteh.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | brcnv 5753 | . 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 2114 Vcvv 3494 class class class wbr 5066 ◡ccnv 5554 ≤ cle 10676 ≥ cge-real 44839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-gte 44841 |
This theorem is referenced by: ex-gte 44848 |
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