Theorem bj-intexr 15400

Description: intexr 4179 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 15388	. . 3 |- -. _V e. _V
2		inteq 3873	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3884	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2242	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2262	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 676	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2418	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   (/)c0 3446   |^|cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-bdn 15309  ax-bdel 15313  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-v 2762  df-dif 3155  df-nul 3447  df-int 3871

This theorem is referenced by: (None)