Theorem cnvpsb 18594

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

Step	Hyp	Ref	Expression
1		cnvps 18593	. 2 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
2		cnvps 18593	. . 3 ⊢ (◡𝑅 ∈ PosetRel → ◡◡𝑅 ∈ PosetRel)
3		dfrel2 6171	. . . 4 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
4		eleq1 2849	. . . . 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))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ^◡ccnv 5644  Rel wrel 5650  PosetRelcps 18579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ps 18581

This theorem is referenced by: (None)