Theorem ordtri2or 2328

Description: A trichotomy law for ordinal classes.

Assertion

Ref	Expression
ordtri2or	|- ((Ord A /\ Ord B) -> (A e. B \/ B (_ A))

Proof of Theorem ordtri2or

Step	Hyp	Ref	Expression
1		ordtri3or 2230	. . 3 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))
2		3orass 584	. . 3 |- ((A e. B \/ A = B \/ B e. A) <-> (A e. B \/ (A = B \/ B e. A)))
3	1, 2	sylib 173	. 2 |- ((Ord A /\ Ord B) -> (A e. B \/ (A = B \/ B e. A)))
4		ordsseleq 2227	. . . . 5 |- ((Ord B /\ Ord A) -> (B (_ A <-> (B e. A \/ B = A)))
5	4	ancoms 334	. . . 4 |- ((Ord A /\ Ord B) -> (B (_ A <-> (B e. A \/ B = A)))
6		orcom 209	. . . . 5 |- ((B e. A \/ B = A) <-> (B = A \/ B e. A))
7		cleqcom 1103	. . . . . 6 |- (B = A <-> A = B)
8	7	orbi1i 215	. . . . 5 |- ((B = A \/ B e. A) <-> (A = B \/ B e. A))
9	6, 8	bitr 151	. . . 4 |- ((B e. A \/ B = A) <-> (A = B \/ B e. A))
10	5, 9	syl6bb 414	. . 3 |- ((Ord A /\ Ord B) -> (B (_ A <-> (A = B \/ B e. A)))
11	10	orbi2d 466	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ B (_ A) <-> (A e. B \/ (A = B \/ B e. A))))
12	3, 11	mpbird 171	1 |- ((Ord A /\ Ord B) -> (A e. B \/ B (_ A))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198

This theorem is referenced by:  ordtri2or2 2329  onun 2358  oaass 3163  iscard3 3693

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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