Theorem bj-intexr 15554

Description: intexr 4183 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-intexr	|- ( |^| A e. _V -> A =/= (/) )

Proof of Theorem bj-intexr

Step	Hyp	Ref	Expression
1		bj-vprc 15542	. . 3 |- -. _V e. _V
2		inteq 3877	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3888	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2245	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2265	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 676	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2421	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   =/= wne 2367   _Vcvv 2763   (/)c0 3450   |^|cint 3874

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-bdn 15463  ax-bdel 15467  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-nul 3451  df-int 3875

This theorem is referenced by: (None)