Theorem pocnv 33730

Description: The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
pocnv	⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Proof of Theorem pocnv

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poirr 5515	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
2		vex 3436	. . . 4 ⊢ 𝑥 ∈ V
3	2, 2	brcnv 5791	. . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥)
4	1, 3	sylnibr 329	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥)
5		3anrev 1100	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6		potr 5516	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
7	5, 6	sylan2b 594	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
9	2, 8	brcnv 5791	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		vex 3436	. . . . 5 ⊢ 𝑧 ∈ V
11	8, 10	brcnv 5791	. . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	9, 11	anbi12ci 628	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥))
13	2, 10	brcnv 5791	. . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥)
14	7, 12, 13	3imtr4g 296	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧))
15	4, 14	ispod 5512	1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2106   class class class wbr 5074   Po wpo 5501  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-po 5503  df-cnv 5597

This theorem is referenced by: (None)