Theorem 0nelelxp 5145

Description: A member of a Cartesian 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 5131	. 2 |- ( C e. ( A X. B ) <-> E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
2		0nelop 4960	. . . . 5 |- -. (/) e. <. x , y >.
3		eleq2 2690	. . . . 5 |- ( C = <. x , y >. -> ( (/) e. C <-> (/) e. <. x , y >. ) )
4	2, 3	mtbiri 317	. . . 4 |- ( C = <. x , y >. -> -. (/) e. C )
5	4	adantr 481	. . 3 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
6	5	exlimivv 1860	. 2 |- ( E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
7	1, 6	sylbi 207	1 |- ( C e. ( A X. B ) -> -. (/) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   <.cop 4183   X. cxp 5112

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-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-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  dmsn0el  5604  onxpdisj  5847