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Mirrors > Home > MPE Home > Th. List > cnvpsb | Structured version Visualization version GIF version |
Description: The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.) |
Ref | Expression |
---|---|
cnvpsb | ⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvps 17810 | . 2 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
2 | cnvps 17810 | . . 3 ⊢ (◡𝑅 ∈ PosetRel → ◡◡𝑅 ∈ PosetRel) | |
3 | dfrel2 6039 | . . . 4 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
4 | eleq1 2897 | . . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel)) | |
5 | 4 | biimpd 230 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
6 | 3, 5 | sylbi 218 | . . 3 ⊢ (Rel 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
7 | 2, 6 | syl5 34 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
8 | 1, 7 | impbid2 227 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 Rel wrel 5553 PosetRelcps 17796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ps 17798 |
This theorem is referenced by: (None) |
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