Theorem bj-intexr 10699

Description: intexr 3925 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 10687	. . 3 |- -. _V e. _V
2		inteq 3639	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3650	. . . . 5 |- |^| (/) = _V
4	2, 3	syl6eq 2129	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2147	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 632	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2299	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   =/= wne 2245   _Vcvv 2601   (/)c0 3251   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdn 10608  ax-bdel 10612  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-nul 3252  df-int 3637

This theorem is referenced by: (None)