Theorem op1stb 4575

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 3861	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 3932	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4275	. . . . . 6 ⊢ {𝐴} ∈ V
6		prexg 4301	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	intpr 3960	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9		snsspr1 3821	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		df-ss 3213	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
11	9, 10	mpbi 145	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
12	8, 11	eqtri 2252	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
13	4, 12	eqtri 2252	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
14	13	inteqi 3932	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
15	1	intsn 3963	. 2 ⊢ ∩ {𝐴} = 𝐴
16	14, 15	eqtri 2252	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∩ cin 3199   ⊆ wss 3200  {csn 3669  {cpr 3670  ⟨cop 3672  ∩ cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-int 3929

This theorem is referenced by:  elreldm  4958  op2ndb  5220  1stval2  6317  fundmen  6980  xpsnen  7004