Theorem 0nelxpOLD 5144

Description: Obsolete proof of 0nelxp 5143 as of 13-Aug-2021. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
0nelxpOLD	|- -. (/) e. ( A X. B )

Proof of Theorem 0nelxpOLD

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

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2		vex 3203	. . . . . 6 |- y e. _V
3	1, 2	opnzi 4943	. . . . 5 |- <. x , y >. =/= (/)
4		simpl 473	. . . . . . 7 |- ( ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> (/) = <. x , y >. )
5	4	eqcomd 2628	. . . . . 6 |- ( ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> <. x , y >. = (/) )
6	5	necon3ai 2819	. . . . 5 |- ( <. x , y >. =/= (/) -> -. ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
7	3, 6	ax-mp 5	. . . 4 |- -. ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) )
8	7	nex 1731	. . 3 |- -. E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) )
9	8	nex 1731	. 2 |- -. E. x E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) )
10		elxp 5131	. 2 |- ( (/) e. ( A X. B ) <-> E. x E. y ( (/) = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
11	9, 10	mtbir 313	1 |- -. (/) e. ( A X. B )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)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: (None)