Theorem cnvso 6291

Description: The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)

Assertion

Ref	Expression
cnvso	⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)

Proof of Theorem cnvso

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

Step	Hyp	Ref	Expression
1		cnvpo 6290	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)
2		ralcom 3293	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5870	. . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2048	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5870	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1174	. . . . 5 ⊢ ((𝑦◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	8	2ralbii 3140	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥◡𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
10	2, 9	bitr4i 281	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥◡𝑅𝑦))
11	1, 10	anbi12i 639	. 2 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (◡𝑅 Po 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥◡𝑅𝑦)))
12		df-so 5572	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5572	. 2 ⊢ (◡𝑅 Or 𝐴 ↔ (◡𝑅 Po 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦◡𝑅𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥◡𝑅𝑦)))
14	11, 12, 13	3bitr4i 306	1 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ w3o 1102  ∀wral 3079   class class class wbr 5110   Po wpo 5569   Or wor 5570  ^◡ccnv 5662

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-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-po 5571  df-so 5572  df-cnv 5671

This theorem is referenced by:  infexd  9445  eqinf  9446  infval  9448  infcl  9450  inflb  9451  infglb  9452  infglbb  9453  fiinfcl  9464  infltoreq  9465  infempty  9470  infiso  9471  wofib  9508  oemapso  9652  cflim2  10248  fin23lem40  10336  gtso  11292  nomaxmo  27840  tosglb  33273  xrsclat  33309  xrge0iifiso  34303  socnv  36234  welb  38365  xrgtso  46041