Theorem 0nelelxp 4754

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 4742	. 2 |- ( C e. ( A X. B ) <-> E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
2		0nelop 4340	. . . 4 |- -. (/) e. <. x , y >.
3		simpl 109	. . . . 5 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> C = <. x , y >. )
4	3	eleq2d 2301	. . . 4 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> ( (/) e. C <-> (/) e. <. x , y >. ) )
5	2, 4	mtbiri 681	. . 3 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
6	5	exlimivv 1945	. 2 |- ( E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
7	1, 6	sylbi 121	1 |- ( C e. ( A X. B ) -> -. (/) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   (/)c0 3494   <.cop 3672   X. cxp 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731

This theorem is referenced by:  dmsn0el  5206