Theorem bj-intexr 16848

Description: intexr 4281 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 16836	. . 3 |- -. _V e. _V
2		inteq 3968	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3979	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2287	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2307	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 686	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2474	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   |^|cint 3965

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-bdn 16757  ax-bdel 16761  ax-bdsep 16824

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-nul 3521  df-int 3966

This theorem is referenced by: (None)