Theorem ordtri2 5758

Description: A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)

Assertion

Ref	Expression
ordtri2	|- ( ( Ord A /\ Ord B ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )

Proof of Theorem ordtri2

Step	Hyp	Ref	Expression
1		ordsseleq 5752	. . . . 5 |- ( ( Ord B /\ Ord A ) -> ( B C_ A <-> ( B e. A \/ B = A ) ) )
2		eqcom 2629	. . . . . . 7 |- ( B = A <-> A = B )
3	2	orbi2i 541	. . . . . 6 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
4		orcom 402	. . . . . 6 |- ( ( B e. A \/ A = B ) <-> ( A = B \/ B e. A ) )
5	3, 4	bitri 264	. . . . 5 |- ( ( B e. A \/ B = A ) <-> ( A = B \/ B e. A ) )
6	1, 5	syl6bb 276	. . . 4 |- ( ( Ord B /\ Ord A ) -> ( B C_ A <-> ( A = B \/ B e. A ) ) )
7		ordtri1 5756	. . . 4 |- ( ( Ord B /\ Ord A ) -> ( B C_ A <-> -. A e. B ) )
8	6, 7	bitr3d 270	. . 3 |- ( ( Ord B /\ Ord A ) -> ( ( A = B \/ B e. A ) <-> -. A e. B ) )
9	8	ancoms 469	. 2 |- ( ( Ord A /\ Ord B ) -> ( ( A = B \/ B e. A ) <-> -. A e. B ) )
10	9	con2bid 344	1 |- ( ( Ord A /\ Ord B ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  ordtri3  5759  ord0eln0  5779  oaord  7627  omord2  7647  oeord  7668  nnaord  7699  nnmord  7712  noextenddif  31821