Theorem eliincex 39293

Description: Counterexample to show that the additional conditions in eliin 4525 and eliin2 39299 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
eliinct.1	|- A = _V
eliinct.2	|- B = (/)

Assertion

Ref	Expression
eliincex	|- -. ( A e. |^|_ x e. B C <-> A. x e. B A e. C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem eliincex

Step	Hyp	Ref	Expression
1		eliinct.1	. . 3 |- A = _V
2		nvel 4797	. . 3 |- -. _V e. |^|_ x e. B C
3	1, 2	eqneltri 39246	. 2 |- -. A e. |^|_ x e. B C
4		ral0 4076	. . 3 |- A. x e. (/) A e. C
5		eliinct.2	. . . 4 |- B = (/)
6	5	raleqi 3142	. . 3 |- ( A. x e. B A e. C <-> A. x e. (/) A e. C )
7	4, 6	mpbir 221	. 2 |- A. x e. B A e. C
8		pm3.22 465	. . . 4 |- ( ( -. A e. |^|_ x e. B C /\ A. x e. B A e. C ) -> ( A. x e. B A e. C /\ -. A e. |^|_ x e. B C ) )
9	8	olcd 408	. . 3 |- ( ( -. A e. |^|_ x e. B C /\ A. x e. B A e. C ) -> ( ( A e. |^|_ x e. B C /\ -. A. x e. B A e. C ) \/ ( A. x e. B A e. C /\ -. A e. |^|_ x e. B C ) ) )
10		xor 935	. . 3 |- ( -. ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) <-> ( ( A e. |^|_ x e. B C /\ -. A. x e. B A e. C ) \/ ( A. x e. B A e. C /\ -. A e. |^|_ x e. B C ) ) )
11	9, 10	sylibr 224	. 2 |- ( ( -. A e. |^|_ x e. B C /\ A. x e. B A e. C ) -> -. ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
12	3, 7, 11	mp2an 708	1 |- -. ( A e. |^|_ x e. B C <-> A. x e. B A e. C )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   |^|_ciin 4521

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

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-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)