Theorem ordtri2or 3074

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 2976	. . 3 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))
2		3orass 777	. . 3 |- ((A e. B \/ A = B \/ B e. A) <-> (A e. B \/ (A = B \/ B e. A)))
3	1, 2	sylib 198	. 2 |- ((Ord A /\ Ord B) -> (A e. B \/ (A = B \/ B e. A)))
4		ordsseleq 2973	. . . . 5 |- ((Ord B /\ Ord A) -> (B (_ A <-> (B e. A \/ B = A)))
5	4	ancoms 436	. . . 4 |- ((Ord A /\ Ord B) -> (B (_ A <-> (B e. A \/ B = A)))
6		orcom 246	. . . . 5 |- ((B e. A \/ B = A) <-> (B = A \/ B e. A))
7		eqcom 1476	. . . . . 6 |- (B = A <-> A = B)
8	7	orbi1i 256	. . . . 5 |- ((B = A \/ B e. A) <-> (A = B \/ B e. A))
9	6, 8	bitr 173	. . . 4 |- ((B e. A \/ B = A) <-> (A = B \/ B e. A))
10	5, 9	syl6bb 535	. . 3 |- ((Ord A /\ Ord B) -> (B (_ A <-> (A = B \/ B e. A)))
11	10	orbi2d 613	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ B (_ A) <-> (A e. B \/ (A = B \/ B e. A))))
12	3, 11	mpbird 196	1 |- ((Ord A /\ Ord B) -> (A e. B \/ B (_ A))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 773   = wceq 955   e. wcel 957   (_ wss 2045  Ord word 2944

This theorem is referenced by:  ordtri2or2 3075  onun 3107  ordunisuc2 3112  oaass 4192  iscard3 4875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948

Copyright terms: Public domain