Theorem ordtri2or 3067

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 2969	. . 3 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))
2		3orass 776	. . 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 2966	. . . . 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 1469	. . . . . 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 534	. . 3 |- ((Ord A /\ Ord B) -> (B (_ A <-> (A = B \/ B e. A)))
11	10	orbi2d 612	. 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 772   = wceq 953   e. wcel 955   (_ wss 2037  Ord word 2937

This theorem is referenced by:  ordtri2or2 3068  onun 3100  ordunisuc2 3105  oaass 4179  iscard3 4860

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941

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