Theorem intprd 10367

Description: The intersection of a pair is the intersection of its members. Closed for of intpr 2553.Theorem 71 of [Suppes] p. 42.

Assertion

Ref	Expression
intprd	⊢ ((A ∈ V ⋀ B ∈ V) → ∩{A, B} = (A ∩ B))

Proof of Theorem intprd

Step	Hyp	Ref	Expression
1		preq1 2438	. . . 4 ⊢ (A = if(A ∈ V, A, ∅) → {A, B} = { if(A ∈ V, A, ∅), B})
2	1	inteqd 2528	. . 3 ⊢ (A = if(A ∈ V, A, ∅) → ∩{A, B} = ∩{ if(A ∈ V, A, ∅), B})
3		ineq1 2200	. . 3 ⊢ (A = if(A ∈ V, A, ∅) → (A ∩ B) = ( if(A ∈ V, A, ∅) ∩ B))
4	2, 3	eqeq12d 1481	. 2 ⊢ (A = if(A ∈ V, A, ∅) → (∩{A, B} = (A ∩ B) ↔ ∩{ if(A ∈ V, A, ∅), B} = ( if(A ∈ V, A, ∅) ∩ B)))
5		preq2 2439	. . . 4 ⊢ (B = if(B ∈ V, B, ∅) → { if(A ∈ V, A, ∅), B} = { if(A ∈ V, A, ∅), if(B ∈ V, B, ∅)})
6	5	inteqd 2528	. . 3 ⊢ (B = if(B ∈ V, B, ∅) → ∩{ if(A ∈ V, A, ∅), B} = ∩{ if(A ∈ V, A, ∅), if(B ∈ V, B, ∅)})
7		ineq2 2201	. . 3 ⊢ (B = if(B ∈ V, B, ∅) → ( if(A ∈ V, A, ∅) ∩ B) = ( if(A ∈ V, A, ∅) ∩ if(B ∈ V, B, ∅)))
8	6, 7	eqeq12d 1481	. 2 ⊢ (B = if(B ∈ V, B, ∅) → (∩{ if(A ∈ V, A, ∅), B} = ( if(A ∈ V, A, ∅) ∩ B) ↔ ∩{ if(A ∈ V, A, ∅), if(B ∈ V, B, ∅)} = ( if(A ∈ V, A, ∅) ∩ if(B ∈ V, B, ∅))))
9		0ex 2701	. . . 4 ⊢ ∅ ∈ V
10	9	elimel 2384	. . 3 ⊢ if(A ∈ V, A, ∅) ∈ V
11	9	elimel 2384	. . 3 ⊢ if(B ∈ V, B, ∅) ∈ V
12	10, 11	intpr 2553	. 2 ⊢ ∩{ if(A ∈ V, A, ∅), if(B ∈ V, B, ∅)} = ( if(A ∈ V, A, ∅) ∩ if(B ∈ V, B, ∅))
13	4, 8, 12	dedth2h 2377	1 ⊢ ((A ∈ V ⋀ B ∈ V) → ∩{A, B} = (A ∩ B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955  Vcvv 1802   ∩ cin 2036  ∅c0 2270   ifcif 2351  {cpr 2400  ∩cint 2523

This theorem is referenced by:  cnfilca 10451

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-11 964  ax-12 965  ax-14 967  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  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-nul 2271  df-if 2352  df-sn 2402  df-pr 2403  df-int 2524

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