Theorem zfregOLD 8502

Description: Obsolete version of zfreg 8500 as of 28-Apr-2021. (Contributed by NM, 26-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
zfregOLD.1	|- A e. _V

Assertion

Ref	Expression
zfregOLD	|- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Distinct variable group:   x, A

Proof of Theorem zfregOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfregOLD.1	. . 3 |- A e. _V
2	1	zfregclOLD 8501	. 2 |- ( E. x x e. A -> E. x e. A A. y e. x -. y e. A )
3		n0 3931	. 2 |- ( A =/= (/) <-> E. x x e. A )
4		disj 4017	. . 3 |- ( ( x i^i A ) = (/) <-> A. y e. x -. y e. A )
5	4	rexbii 3041	. 2 |- ( E. x e. A ( x i^i A ) = (/) <-> E. x e. A A. y e. x -. y e. A )
6	2, 3, 5	3imtr4i 281	1 |- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   (/)c0 3915

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-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by: (None)