Theorem prtlem14 34159

Description: Lemma for prter1 34164, prter2 34166 and prtex 34165. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
prtlem14	|- ( Prt A -> ( ( x e. A /\ y e. A ) -> ( ( w e. x /\ w e. y ) -> x = y ) ) )

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

Allowed substitution hint:   A( w)

Proof of Theorem prtlem14

Step	Hyp	Ref	Expression
1		df-prt 34157	. . 3 |- ( Prt A <-> A. x e. A A. y e. A ( x = y \/ ( x i^i y ) = (/) ) )
2		rsp2 2936	. . 3 |- ( A. x e. A A. y e. A ( x = y \/ ( x i^i y ) = (/) ) -> ( ( x e. A /\ y e. A ) -> ( x = y \/ ( x i^i y ) = (/) ) ) )
3	1, 2	sylbi 207	. 2 |- ( Prt A -> ( ( x e. A /\ y e. A ) -> ( x = y \/ ( x i^i y ) = (/) ) ) )
4		elin 3796	. . . 4 |- ( w e. ( x i^i y ) <-> ( w e. x /\ w e. y ) )
5		eq0 3929	. . . . . 6 |- ( ( x i^i y ) = (/) <-> A. w -. w e. ( x i^i y ) )
6		sp 2053	. . . . . 6 |- ( A. w -. w e. ( x i^i y ) -> -. w e. ( x i^i y ) )
7	5, 6	sylbi 207	. . . . 5 |- ( ( x i^i y ) = (/) -> -. w e. ( x i^i y ) )
8	7	pm2.21d 118	. . . 4 |- ( ( x i^i y ) = (/) -> ( w e. ( x i^i y ) -> x = y ) )
9	4, 8	syl5bir 233	. . 3 |- ( ( x i^i y ) = (/) -> ( ( w e. x /\ w e. y ) -> x = y ) )
10	9	jao1i 825	. 2 |- ( ( x = y \/ ( x i^i y ) = (/) ) -> ( ( w e. x /\ w e. y ) -> x = y ) )
11	3, 10	syl6 35	1 |- ( Prt A -> ( ( x e. A /\ y e. A ) -> ( ( w e. x /\ w e. y ) -> x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   (/)c0 3915   Prt wprt 34156

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-prt 34157

This theorem is referenced by:  prtlem15  34160  prtlem17  34161