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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 17035 | . 2 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
2 | cnvps 17035 | . . 3 ⊢ (◡𝑅 ∈ PosetRel → ◡◡𝑅 ∈ PosetRel) | |
3 | dfrel2 5502 | . . . 4 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
4 | eleq1 2676 | . . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel)) | |
5 | 4 | biimpd 218 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
6 | 3, 5 | sylbi 206 | . . 3 ⊢ (Rel 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
7 | 2, 6 | syl5 33 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
8 | 1, 7 | impbid2 215 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ◡ccnv 5037 Rel wrel 5043 PosetRelcps 17021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ps 17023 |
This theorem is referenced by: (None) |
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