Theorem ordtri2or 3077

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 2979	. . 3 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))
2		3orass 778	. . 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 2976	. . . . 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 1477	. . . . . 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 536	. . 3 |- ((Ord A /\ Ord B) -> (B (_ A <-> (A = B \/ B e. A)))
11	10	orbi2d 614	. 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 774   = wceq 956   e. wcel 958   (_ wss 2047  Ord word 2947

This theorem is referenced by:  ordtri2or2 3078  onun 3110  ordunisuc2 3115  oaass 4195  iscard3 4888

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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