Theorem ordelinel 5825

Description: The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.)

Assertion

Ref	Expression
ordelinel	|- ( ( Ord A /\ Ord B /\ Ord C ) -> ( ( A i^i B ) e. C <-> ( A e. C \/ B e. C ) ) )

Proof of Theorem ordelinel

Step	Hyp	Ref	Expression
1		ordtri2or3 5824	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( A = ( A i^i B ) \/ B = ( A i^i B ) ) )
2	1	3adant3 1081	. . 3 |- ( ( Ord A /\ Ord B /\ Ord C ) -> ( A = ( A i^i B ) \/ B = ( A i^i B ) ) )
3		eleq1a 2696	. . . 4 |- ( ( A i^i B ) e. C -> ( A = ( A i^i B ) -> A e. C ) )
4		eleq1a 2696	. . . 4 |- ( ( A i^i B ) e. C -> ( B = ( A i^i B ) -> B e. C ) )
5	3, 4	orim12d 883	. . 3 |- ( ( A i^i B ) e. C -> ( ( A = ( A i^i B ) \/ B = ( A i^i B ) ) -> ( A e. C \/ B e. C ) ) )
6	2, 5	syl5com 31	. 2 |- ( ( Ord A /\ Ord B /\ Ord C ) -> ( ( A i^i B ) e. C -> ( A e. C \/ B e. C ) ) )
7		ordin 5753	. . 3 |- ( ( Ord A /\ Ord B ) -> Ord ( A i^i B ) )
8		inss1 3833	. . . . 5 |- ( A i^i B ) C_ A
9		ordtr2 5768	. . . . 5 |- ( ( Ord ( A i^i B ) /\ Ord C ) -> ( ( ( A i^i B ) C_ A /\ A e. C ) -> ( A i^i B ) e. C ) )
10	8, 9	mpani 712	. . . 4 |- ( ( Ord ( A i^i B ) /\ Ord C ) -> ( A e. C -> ( A i^i B ) e. C ) )
11		inss2 3834	. . . . 5 |- ( A i^i B ) C_ B
12		ordtr2 5768	. . . . 5 |- ( ( Ord ( A i^i B ) /\ Ord C ) -> ( ( ( A i^i B ) C_ B /\ B e. C ) -> ( A i^i B ) e. C ) )
13	11, 12	mpani 712	. . . 4 |- ( ( Ord ( A i^i B ) /\ Ord C ) -> ( B e. C -> ( A i^i B ) e. C ) )
14	10, 13	jaod 395	. . 3 |- ( ( Ord ( A i^i B ) /\ Ord C ) -> ( ( A e. C \/ B e. C ) -> ( A i^i B ) e. C ) )
15	7, 14	stoic3 1701	. 2 |- ( ( Ord A /\ Ord B /\ Ord C ) -> ( ( A e. C \/ B e. C ) -> ( A i^i B ) e. C ) )
16	6, 15	impbid 202	1 |- ( ( Ord A /\ Ord B /\ Ord C ) -> ( ( A i^i B ) e. C <-> ( A e. C \/ B e. C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   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:  mreexexdOLD  16309