Theorem trficl 37961

Description: The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)

Hypothesis

Ref	Expression
trficl.a	|- A = { z | ( z o. z ) C_ z }

Assertion

Ref	Expression
trficl	|- A. x e. A A. y e. A ( x i^i y ) e. A

Distinct variable groups:   x, y, z   y, A

Allowed substitution hints:   A( x, z)

Proof of Theorem trficl

Step	Hyp	Ref	Expression
1		trficl.a	. 2 |- A = { z | ( z o. z ) C_ z }
2		vex 3203	. . 3 |- x e. _V
3	2	inex1 4799	. 2 |- ( x i^i y ) e. _V
4		id 22	. . . 4 |- ( z = ( x i^i y ) -> z = ( x i^i y ) )
5	4, 4	coeq12d 5286	. . 3 |- ( z = ( x i^i y ) -> ( z o. z ) = ( ( x i^i y ) o. ( x i^i y ) ) )
6	5, 4	sseq12d 3634	. 2 |- ( z = ( x i^i y ) -> ( ( z o. z ) C_ z <-> ( ( x i^i y ) o. ( x i^i y ) ) C_ ( x i^i y ) ) )
7		id 22	. . . 4 |- ( z = x -> z = x )
8	7, 7	coeq12d 5286	. . 3 |- ( z = x -> ( z o. z ) = ( x o. x ) )
9	8, 7	sseq12d 3634	. 2 |- ( z = x -> ( ( z o. z ) C_ z <-> ( x o. x ) C_ x ) )
10		id 22	. . . 4 |- ( z = y -> z = y )
11	10, 10	coeq12d 5286	. . 3 |- ( z = y -> ( z o. z ) = ( y o. y ) )
12	11, 10	sseq12d 3634	. 2 |- ( z = y -> ( ( z o. z ) C_ z <-> ( y o. y ) C_ y ) )
13		trin2 5519	. 2 |- ( ( ( x o. x ) C_ x /\ ( y o. y ) C_ y ) -> ( ( x i^i y ) o. ( x i^i y ) ) C_ ( x i^i y ) )
14	1, 3, 6, 9, 12, 13	cllem0 37871	1 |- A. x e. A A. y e. A ( x i^i y ) e. A

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   o. ccom 5118

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-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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by: (None)