Theorem pocnv 32167

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 5244	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
2		vex 3388	. . . 4 ⊢ 𝑥 ∈ V
3	2, 2	brcnv 5508	. . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥)
4	1, 3	sylnibr 321	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥)
5		3anrev 1127	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6		potr 5245	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
7	5, 6	sylan2b 588	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8		vex 3388	. . . . 5 ⊢ 𝑦 ∈ V
9	2, 8	brcnv 5508	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		vex 3388	. . . . 5 ⊢ 𝑧 ∈ V
11	8, 10	brcnv 5508	. . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	9, 11	anbi12ci 622	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥))
13	2, 10	brcnv 5508	. . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥)
14	7, 12, 13	3imtr4g 288	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧))
15	4, 14	ispod 5241	1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   ∈ wcel 2157   class class class wbr 4843   Po wpo 5231  ^◡ccnv 5311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-po 5233  df-cnv 5320

This theorem is referenced by: (None)