Theorem intnex 2726

Description: If a class intersection is not a set, it must be the universe.

Assertion

Ref	Expression
intnex	|- (-. |^|A e. V <-> |^|A = V)

Proof of Theorem intnex

Step	Hyp	Ref	Expression
1		intex 2725	. . . 4 |- (A =/= (/) <-> |^|A e. V)
2	1	necon1bbii 1615	. . 3 |- (-. |^|A e. V <-> A = (/))
3		inteq 2532	. . . 4 |- (A = (/) -> |^|A = |^|(/))
4		int0 2543	. . . 4 |- |^|(/) = V
5	3, 4	syl6eq 1521	. . 3 |- (A = (/) -> |^|A = V)
6	2, 5	sylbi 199	. 2 |- (-. |^|A e. V -> |^|A = V)
7		nvelv 2709	. . 3 |- -. V e. V
8		eleq1 1532	. . 3 |- (|^|A = V -> (|^|A e. V <-> V e. V))
9	7, 8	mtbiri 716	. 2 |- (|^|A = V -> -. |^|A e. V)
10	6, 9	impbi 157	1 |- (-. |^|A e. V <-> |^|A = V)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 955   e. wcel 957  Vcvv 1808  (/)c0 2277  |^|cint 2529

This theorem is referenced by:  intabs 2729

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278  df-int 2530

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