Theorem bj-intexr 16229

Description: intexr 4233 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 16217	. . 3 |- -. _V e. _V
2		inteq 3925	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3936	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2278	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2298	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 679	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2454	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   |^|cint 3922

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-bdn 16138  ax-bdel 16142  ax-bdsep 16205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-nul 3492  df-int 3923

This theorem is referenced by: (None)