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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 17816 | . 2 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) | |
2 | cnvps 17816 | . . 3 ⊢ (◡𝑅 ∈ PosetRel → ◡◡𝑅 ∈ PosetRel) | |
3 | dfrel2 6040 | . . . 4 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
4 | eleq1 2900 | . . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel)) | |
5 | 4 | biimpd 231 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
6 | 3, 5 | sylbi 219 | . . 3 ⊢ (Rel 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
7 | 2, 6 | syl5 34 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)) |
8 | 1, 7 | impbid2 228 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ◡ccnv 5548 Rel wrel 5554 PosetRelcps 17802 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ps 17804 |
This theorem is referenced by: (None) |
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