Theorem cnvso 6279

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 6278	. . 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 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
4		vex 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5859	. . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
6		equcom 2041	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	4, 3	brcnv 5859	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 6, 7	3orbi123i 1172	. . . . 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 5561	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
13		df-so 5561	. 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 1100  ∀wral 3079   class class class wbr 5105   Po wpo 5558   Or wor 5559  ^◡ccnv 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-po 5560  df-so 5561  df-cnv 5660

This theorem is referenced by:  infexd  9432  eqinf  9433  infval  9435  infcl  9437  inflb  9438  infglb  9439  infglbb  9440  fiinfcl  9451  infltoreq  9452  infempty  9457  infiso  9458  wofib  9495  oemapso  9639  cflim2  10235  fin23lem40  10323  gtso  11279  nomaxmo  27820  tosglb  33208  xrsclat  33244  xrge0iifiso  34242  socnv  36127  welb  38247  xrgtso  45919