Theorem prtlem12 34152

Description: Lemma for prtex 34165 and prter3 34167. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
prtlem12	|- ( .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) } -> Rel .~ )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y, u)   .~ ( x, y, u)

Proof of Theorem prtlem12

Step	Hyp	Ref	Expression
1		relopab 5247	. 2 |- Rel { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
2		releq 5201	. 2 |- ( .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) } -> ( Rel .~ <-> Rel { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) } ) )
3	1, 2	mpbiri 248	1 |- ( .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) } -> Rel .~ )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wrex 2913   {copab 4712   Rel wrel 5119

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-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-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-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)