Theorem sopc 5934

Description: Linear ordering as partial, connected relationship. (Contributed by SF, 12-Mar-2015.)

Assertion

Ref	Expression
sopc	|- (R Or A <-> (R Po A /\ R Connex A))

Proof of Theorem sopc

Step	Hyp	Ref	Expression
1		df-strict 5904	. . 3 |- Or = ( Po i^i Connex )
2	1	breqi 4645	. 2 |- (R Or A <-> R( Po i^i Connex )A)
3		brin 4693	. 2 |- (R( Po i^i Connex )A <-> (R Po A /\ R Connex A))
4	2, 3	bitri 240	1 |- (R Or A <-> (R Po A /\ R Connex A))

Syntax hints:   <-> wb 176   /\ wa 358   i^i cin 3208   class class class wbr 4639   Po cpartial 5891   Connex cconnex 5892   Or cstrict 5893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-br 4640  df-strict 5904

This theorem is referenced by:  sod  5937  weds  5938  so0  5941  nchoicelem8  6296  nchoicelem19  6307