Theorem op1stb 4175

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)

Hypotheses

Ref	Expression
op1stb.1	⊢ A ∈ V
op1stb.2	⊢ B ∈ V

Assertion

Ref	Expression
op1stb	⊢ ∩ ∩ ⟨A, B⟩ = A

Proof of Theorem op1stb

Step	Hyp	Ref	Expression
1		op1stb.1	. . . . . 6 ⊢ A ∈ V
2		op1stb.2	. . . . . 6 ⊢ B ∈ V
3	1, 2	dfop 3539	. . . . 5 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
4	3	inteqi 3610	. . . 4 ⊢ ∩ ⟨A, B⟩ = ∩ {{A}, {A, B}}
5		snexgOLD 3926	. . . . . . 7 ⊢ (A ∈ V → {A} ∈ V)
6	1, 5	ax-mp 7	. . . . . 6 ⊢ {A} ∈ V
7		prexgOLD 3937	. . . . . . 7 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
8	1, 2, 7	mp2an 402	. . . . . 6 ⊢ {A, B} ∈ V
9	6, 8	intpr 3638	. . . . 5 ⊢ ∩ {{A}, {A, B}} = ({A} ∩ {A, B})
10		snsspr1 3503	. . . . . 6 ⊢ {A} ⊆ {A, B}
11		df-ss 2925	. . . . . 6 ⊢ ({A} ⊆ {A, B} ↔ ({A} ∩ {A, B}) = {A})
12	10, 11	mpbi 133	. . . . 5 ⊢ ({A} ∩ {A, B}) = {A}
13	9, 12	eqtri 2057	. . . 4 ⊢ ∩ {{A}, {A, B}} = {A}
14	4, 13	eqtri 2057	. . 3 ⊢ ∩ ⟨A, B⟩ = {A}
15	14	inteqi 3610	. 2 ⊢ ∩ ∩ ⟨A, B⟩ = ∩ {A}
16	1	intsn 3641	. 2 ⊢ ∩ {A} = A
17	15, 16	eqtri 2057	1 ⊢ ∩ ∩ ⟨A, B⟩ = A

Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ⊆ wss 2911  {csn 3367  {cpr 3368  ⟨cop 3370  ∩ cint 3606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607

This theorem is referenced by:  elreldm  4503  op2ndb  4747  1stval2  5724  fundmen  6222  xpsnen  6231