Theorem socnv 30908

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

Assertion

Ref	Expression
socnv	⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴)

Proof of Theorem socnv

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

Step	Hyp	Ref	Expression
1		sopo 4976	. . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
2		pocnv 30907	. . 3 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝑅 Or 𝐴 → ◡𝑅 Po 𝐴)
4		solin 4982	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
5		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
6		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 5227	. . . . 5 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		biid 250	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
9	6, 5	brcnv 5227	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
10	7, 8, 9	3orbi123i 1245	. . . 4 ⊢ ((𝑥◡𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦◡𝑅𝑥) ↔ (𝑦𝑅𝑥 ∨ 𝑥 = 𝑦 ∨ 𝑥𝑅𝑦))
11		orcom 401	. . . . . 6 ⊢ ((𝑦𝑅𝑥 ∨ 𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
12	11	orbi2i 540	. . . . 5 ⊢ ((𝑥 = 𝑦 ∨ (𝑦𝑅𝑥 ∨ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
13		3orass 1034	. . . . . 6 ⊢ ((𝑦𝑅𝑥 ∨ 𝑥 = 𝑦 ∨ 𝑥𝑅𝑦) ↔ (𝑦𝑅𝑥 ∨ (𝑥 = 𝑦 ∨ 𝑥𝑅𝑦)))
14		or12 544	. . . . . 6 ⊢ ((𝑦𝑅𝑥 ∨ (𝑥 = 𝑦 ∨ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑦 ∨ (𝑦𝑅𝑥 ∨ 𝑥𝑅𝑦)))
15	13, 14	bitri 263	. . . . 5 ⊢ ((𝑦𝑅𝑥 ∨ 𝑥 = 𝑦 ∨ 𝑥𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ (𝑦𝑅𝑥 ∨ 𝑥𝑅𝑦)))
16		3orass 1034	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
17		or12 544	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
18	16, 17	bitri 263	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
19	12, 15, 18	3bitr4i 291	. . . 4 ⊢ ((𝑦𝑅𝑥 ∨ 𝑥 = 𝑦 ∨ 𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
20	10, 19	bitri 263	. . 3 ⊢ ((𝑥◡𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦◡𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
21	4, 20	sylibr 223	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥◡𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦◡𝑅𝑥))
22	3, 21	issod 4989	1 ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∈ wcel 1977   class class class wbr 4583   Po wpo 4957   Or wor 4958  ^◡ccnv 5037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-cnv 5046

This theorem is referenced by:  wzelOLD  31016