Theorem neiopne 10369

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

Assertion

Ref	Expression
neiopne	⊢ ((A ∩ B) ≠ ∅ → (A ≠ ∅ ⋀ B ≠ ∅))

Proof of Theorem neiopne

Step	Hyp	Ref	Expression
1		ineq1 2200	. . . . 5 ⊢ (A = ∅ → (A ∩ B) = (∅ ∩ B))
2		incom 2198	. . . . . 6 ⊢ (∅ ∩ B) = (B ∩ ∅)
3		eqtrt 1484	. . . . . . 7 ⊢ (((A ∩ B) = (∅ ∩ B) ⋀ (∅ ∩ B) = (B ∩ ∅)) → (A ∩ B) = (B ∩ ∅))
4		in0 2288	. . . . . . 7 ⊢ (B ∩ ∅) = ∅
5	3, 4	syl6eq 1515	. . . . . 6 ⊢ (((A ∩ B) = (∅ ∩ B) ⋀ (∅ ∩ B) = (B ∩ ∅)) → (A ∩ B) = ∅)
6	2, 5	mpan2 694	. . . . 5 ⊢ ((A ∩ B) = (∅ ∩ B) → (A ∩ B) = ∅)
7	1, 6	syl 10	. . . 4 ⊢ (A = ∅ → (A ∩ B) = ∅)
8		ineq2 2201	. . . . 5 ⊢ (B = ∅ → (A ∩ B) = (A ∩ ∅))
9		in0 2288	. . . . 5 ⊢ (A ∩ ∅) = ∅
10	8, 9	syl6eq 1515	. . . 4 ⊢ (B = ∅ → (A ∩ B) = ∅)
11	7, 10	jaoi 341	. . 3 ⊢ ((A = ∅ ⋁ B = ∅) → (A ∩ B) = ∅)
12	11	con3i 98	. 2 ⊢ (¬ (A ∩ B) = ∅ → ¬ (A = ∅ ⋁ B = ∅))
13		df-ne 1579	. 2 ⊢ ((A ∩ B) ≠ ∅ ↔ ¬ (A ∩ B) = ∅)
14		neanior 1631	. 2 ⊢ ((A ≠ ∅ ⋀ B ≠ ∅) ↔ ¬ (A = ∅ ⋁ B = ∅))
15	12, 13, 14	3imtr4 219	1 ⊢ ((A ∩ B) ≠ ∅ → (A ≠ ∅ ⋀ B ≠ ∅))

Syntax hints:  ¬ wn 2   → wi 3   ⋁ wo 222   ⋀ wa 223   = wceq 953   ≠ wne 1577   ∩ cin 2036  ∅c0 2270

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-in 2041  df-nul 2271

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