Theorem neiopne 10427

Description: If an intersection is not empty its operands are not empty.

Assertion

Ref	Expression
neiopne	|- ((A i^i B) =/= (/) -> (A =/= (/) /\ B =/= (/)))

Proof of Theorem neiopne

Step	Hyp	Ref	Expression
1		ineq1 2207	. . . . 5 |- (A = (/) -> (A i^i B) = ((/) i^i B))
2		incom 2205	. . . . . 6 |- ((/) i^i B) = (B i^i (/))
3		eqtrt 1490	. . . . . . 7 |- (((A i^i B) = ((/) i^i B) /\ ((/) i^i B) = (B i^i (/))) -> (A i^i B) = (B i^i (/)))
4		in0 2295	. . . . . . 7 |- (B i^i (/)) = (/)
5	3, 4	syl6eq 1521	. . . . . 6 |- (((A i^i B) = ((/) i^i B) /\ ((/) i^i B) = (B i^i (/))) -> (A i^i B) = (/))
6	2, 5	mpan2 695	. . . . 5 |- ((A i^i B) = ((/) i^i B) -> (A i^i B) = (/))
7	1, 6	syl 10	. . . 4 |- (A = (/) -> (A i^i B) = (/))
8		ineq2 2208	. . . . 5 |- (B = (/) -> (A i^i B) = (A i^i (/)))
9		in0 2295	. . . . 5 |- (A i^i (/)) = (/)
10	8, 9	syl6eq 1521	. . . 4 |- (B = (/) -> (A i^i B) = (/))
11	7, 10	jaoi 341	. . 3 |- ((A = (/) \/ B = (/)) -> (A i^i B) = (/))
12	11	con3i 98	. 2 |- (-. (A i^i B) = (/) -> -. (A = (/) \/ B = (/)))
13		df-ne 1585	. 2 |- ((A i^i B) =/= (/) <-> -. (A i^i B) = (/))
14		neanior 1637	. 2 |- ((A =/= (/) /\ B =/= (/)) <-> -. (A = (/) \/ B = (/)))
15	12, 13, 14	3imtr4 219	1 |- ((A i^i B) =/= (/) -> (A =/= (/) /\ B =/= (/)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 955   =/= wne 1583   i^i cin 2043  (/)c0 2277

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-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

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-v 1809  df-dif 2046  df-in 2048  df-nul 2278

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