Theorem intpr 3906

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Hypotheses

Ref	Expression
intpr.1	⊢ 𝐴 ∈ V
intpr.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
intpr	⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Proof of Theorem intpr

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

Step	Hyp	Ref	Expression
1		19.26 1495	. . . 4 ⊢ (∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
2		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
3	2	elpr 3643	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
4	3	imbi1i 238	. . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦))
5		jaob 711	. . . . . 6 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
6	4, 5	bitri 184	. . . . 5 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
7	6	albii 1484	. . . 4 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
8		intpr.1	. . . . . 6 ⊢ 𝐴 ∈ V
9	8	clel4 2900	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦))
10		intpr.2	. . . . . 6 ⊢ 𝐵 ∈ V
11	10	clel4 2900	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))
12	9, 11	anbi12i 460	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)))
13	1, 7, 12	3bitr4i 212	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14		vex 2766	. . . 4 ⊢ 𝑥 ∈ V
15	14	elint 3880	. . 3 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦))
16		elin 3346	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
17	13, 15, 16	3bitr4i 212	. 2 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴 ∩ 𝐵))
18	17	eqriv 2193	1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156  {cpr 3623  ∩ cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-int 3875

This theorem is referenced by:  intprg  3907  op1stb  4513  onintexmid  4609