Theorem ordtr3 5769

Description: Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.)

Assertion

Ref	Expression
ordtr3	|- ( ( Ord B /\ Ord C ) -> ( A e. B -> ( A e. C \/ C e. B ) ) )

Proof of Theorem ordtr3

Step	Hyp	Ref	Expression
1		nelss 3664	. . . . . 6 |- ( ( A e. B /\ -. A e. C ) -> -. B C_ C )
2	1	adantl 482	. . . . 5 |- ( ( ( Ord B /\ Ord C ) /\ ( A e. B /\ -. A e. C ) ) -> -. B C_ C )
3		ordtri1 5756	. . . . . . 7 |- ( ( Ord B /\ Ord C ) -> ( B C_ C <-> -. C e. B ) )
4	3	con2bid 344	. . . . . 6 |- ( ( Ord B /\ Ord C ) -> ( C e. B <-> -. B C_ C ) )
5	4	adantr 481	. . . . 5 |- ( ( ( Ord B /\ Ord C ) /\ ( A e. B /\ -. A e. C ) ) -> ( C e. B <-> -. B C_ C ) )
6	2, 5	mpbird 247	. . . 4 |- ( ( ( Ord B /\ Ord C ) /\ ( A e. B /\ -. A e. C ) ) -> C e. B )
7	6	expr 643	. . 3 |- ( ( ( Ord B /\ Ord C ) /\ A e. B ) -> ( -. A e. C -> C e. B ) )
8	7	orrd 393	. 2 |- ( ( ( Ord B /\ Ord C ) /\ A e. B ) -> ( A e. C \/ C e. B ) )
9	8	ex 450	1 |- ( ( Ord B /\ Ord C ) -> ( A e. B -> ( A e. C \/ C e. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   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: (None)