Theorem intprg 3879

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3878. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
intprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Proof of Theorem intprg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3671	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	inteqd 3851	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑥, 𝑦} = ∩ {𝐴, 𝑦})
3		ineq1 3331	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
4	2, 3	eqeq12d 2192	. 2 ⊢ (𝑥 = 𝐴 → (∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) ↔ ∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦)))
5		preq2 3672	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	inteqd 3851	. . 3 ⊢ (𝑦 = 𝐵 → ∩ {𝐴, 𝑦} = ∩ {𝐴, 𝐵})
7		ineq2 3332	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
8	6, 7	eqeq12d 2192	. 2 ⊢ (𝑦 = 𝐵 → (∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦) ↔ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)))
9		vex 2742	. . 3 ⊢ 𝑥 ∈ V
10		vex 2742	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	intpr 3878	. 2 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
12	4, 8, 11	vtocl2g 2803	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ∩ cin 3130  {cpr 3595  ∩ cint 3846

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-int 3847

This theorem is referenced by:  intsng  3880  op1stbg  4481  subrgin  13370  lssincl  13477