Theorem ordtri3 5759

Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)

Assertion

Ref	Expression
ordtri3	|- ( ( Ord A /\ Ord B ) -> ( A = B <-> -. ( A e. B \/ B e. A ) ) )

Proof of Theorem ordtri3

Step	Hyp	Ref	Expression
1		ordirr 5741	. . . . . 6 |- ( Ord B -> -. B e. B )
2	1	adantl 482	. . . . 5 |- ( ( Ord A /\ Ord B ) -> -. B e. B )
3		eleq2 2690	. . . . . 6 |- ( A = B -> ( B e. A <-> B e. B ) )
4	3	notbid 308	. . . . 5 |- ( A = B -> ( -. B e. A <-> -. B e. B ) )
5	2, 4	syl5ibrcom 237	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( A = B -> -. B e. A ) )
6	5	pm4.71d 666	. . 3 |- ( ( Ord A /\ Ord B ) -> ( A = B <-> ( A = B /\ -. B e. A ) ) )
7		pm5.61 749	. . . 4 |- ( ( ( A = B \/ B e. A ) /\ -. B e. A ) <-> ( A = B /\ -. B e. A ) )
8		pm4.52 512	. . . 4 |- ( ( ( A = B \/ B e. A ) /\ -. B e. A ) <-> -. ( -. ( A = B \/ B e. A ) \/ B e. A ) )
9	7, 8	bitr3i 266	. . 3 |- ( ( A = B /\ -. B e. A ) <-> -. ( -. ( A = B \/ B e. A ) \/ B e. A ) )
10	6, 9	syl6bb 276	. 2 |- ( ( Ord A /\ Ord B ) -> ( A = B <-> -. ( -. ( A = B \/ B e. A ) \/ B e. A ) ) )
11		ordtri2 5758	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )
12	11	orbi1d 739	. . 3 |- ( ( Ord A /\ Ord B ) -> ( ( A e. B \/ B e. A ) <-> ( -. ( A = B \/ B e. A ) \/ B e. A ) ) )
13	12	notbid 308	. 2 |- ( ( Ord A /\ Ord B ) -> ( -. ( A e. B \/ B e. A ) <-> -. ( -. ( A = B \/ B e. A ) \/ B e. A ) ) )
14	10, 13	bitr4d 271	1 |- ( ( Ord A /\ Ord B ) -> ( A = B <-> -. ( A e. B \/ B e. A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   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:  ordunisuc2  7044  tz7.48lem  7536  oacan  7628  omcan  7649  oecan  7669  omsmo  7734  omopthi  7737  inf3lem6  8530  cantnfp1lem3  8577  infpssrlem5  9129  fin23lem24  9144  isf32lem4  9178  om2uzf1oi  12752