Theorem pocnv 33636

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 5506	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
2		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
3	2, 2	brcnv 5780	. . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥)
4	1, 3	sylnibr 328	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥)
5		3anrev 1099	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6		potr 5507	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
7	5, 6	sylan2b 593	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
9	2, 8	brcnv 5780	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		vex 3426	. . . . 5 ⊢ 𝑧 ∈ V
11	8, 10	brcnv 5780	. . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	9, 11	anbi12ci 627	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥))
13	2, 10	brcnv 5780	. . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥)
14	7, 12, 13	3imtr4g 295	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧))
15	4, 14	ispod 5503	1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108   class class class wbr 5070   Po wpo 5492  ^◡ccnv 5579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-po 5494  df-cnv 5588

This theorem is referenced by: (None)