Theorem en3lplem1 8511

Description: Lemma for en3lp 8513. (Contributed by Alan Sare, 28-Oct-2011.)

Assertion

Ref	Expression
en3lplem1	|- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> ( x i^i { A , B , C } ) =/= (/) ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem en3lplem1

Step	Hyp	Ref	Expression
1		simp3 1063	. . 3 |- ( ( A e. B /\ B e. C /\ C e. A ) -> C e. A )
2		eleq2 2690	. . 3 |- ( x = A -> ( C e. x <-> C e. A ) )
3	1, 2	syl5ibrcom 237	. 2 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> C e. x ) )
4		tpid3g 4305	. . . . 5 |- ( C e. A -> C e. { A , B , C } )
5	4	3ad2ant3 1084	. . . 4 |- ( ( A e. B /\ B e. C /\ C e. A ) -> C e. { A , B , C } )
6		inelcm 4032	. . . 4 |- ( ( C e. x /\ C e. { A , B , C } ) -> ( x i^i { A , B , C } ) =/= (/) )
7	5, 6	sylan2 491	. . 3 |- ( ( C e. x /\ ( A e. B /\ B e. C /\ C e. A ) ) -> ( x i^i { A , B , C } ) =/= (/) )
8	7	expcom 451	. 2 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( C e. x -> ( x i^i { A , B , C } ) =/= (/) ) )
9	3, 8	syld 47	1 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> ( x i^i { A , B , C } ) =/= (/) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   (/)c0 3915   {ctp 4181

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  en3lplem2  8512