Theorem pocnv 35757

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 5561	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
2		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
3	2, 2	brcnv 5849	. . 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 5562	. . . 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 3454	. . . . 5 ⊢ 𝑦 ∈ V
9	2, 8	brcnv 5849	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
11	8, 10	brcnv 5849	. . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	9, 11	anbi12ci 629	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥))
13	2, 10	brcnv 5849	. . 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 5558	1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   class class class wbr 5110   Po wpo 5547  ^◡ccnv 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-po 5549  df-cnv 5649

This theorem is referenced by: (None)