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Theorem cnvpsb 17207
Description: The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
cnvpsb (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))

Proof of Theorem cnvpsb
StepHypRef Expression
1 cnvps 17206 . 2 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
2 cnvps 17206 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
3 dfrel2 5581 . . . 4 (Rel 𝑅𝑅 = 𝑅)
4 eleq1 2688 . . . . 5 (𝑅 = 𝑅 → (𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))
54biimpd 219 . . . 4 (𝑅 = 𝑅 → (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
63, 5sylbi 207 . . 3 (Rel 𝑅 → (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
72, 6syl5 34 . 2 (Rel 𝑅 → (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
81, 7impbid2 216 1 (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  ccnv 5111  Rel wrel 5117  PosetRelcps 17192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ps 17194
This theorem is referenced by: (None)
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