Theorem 0nelelxp 4640

Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Assertion

Ref	Expression
0nelelxp	|- ( C e. ( A X. B ) -> -. (/) e. C )

Proof of Theorem 0nelelxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4628	. 2 |- ( C e. ( A X. B ) <-> E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
2		0nelop 4233	. . . 4 |- -. (/) e. <. x , y >.
3		simpl 108	. . . . 5 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> C = <. x , y >. )
4	3	eleq2d 2240	. . . 4 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> ( (/) e. C <-> (/) e. <. x , y >. ) )
5	2, 4	mtbiri 670	. . 3 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
6	5	exlimivv 1889	. 2 |- ( E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
7	1, 6	sylbi 120	1 |- ( C e. ( A X. B ) -> -. (/) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   (/)c0 3414   <.cop 3586   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617

This theorem is referenced by:  dmsn0el  5080