Theorem pocnv 36255

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 5581	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
2		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
3	2, 2	brcnv 5868	. . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥)
4	1, 3	sylnibr 332	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥)
5		3anrev 1118	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6		potr 5582	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
7	5, 6	sylan2b 605	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥))
8		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
9	2, 8	brcnv 5868	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		vex 3459	. . . . 5 ⊢ 𝑧 ∈ V
11	8, 10	brcnv 5868	. . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	9, 11	anbi12ci 640	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥))
13	2, 10	brcnv 5868	. . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥)
14	7, 12, 13	3imtr4g 299	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧))
15	4, 14	ispod 5578	1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143   class class class wbr 5109   Po wpo 5567  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-po 5569  df-cnv 5669

This theorem is referenced by: (None)