Theorem bj-intexr 16503

Description: intexr 4240 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 16491	. . 3 |- -. _V e. _V
2		inteq 3931	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3942	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2280	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2300	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 681	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2456	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   |^|cint 3928

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-bdn 16412  ax-bdel 16416  ax-bdsep 16479

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-nul 3495  df-int 3929

This theorem is referenced by: (None)