ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  op1stb GIF version

Theorem op1stb 4255
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
StepHypRef Expression
1 op1stb.1 . . . . . 6 𝐴 ∈ V
2 op1stb.2 . . . . . 6 𝐵 ∈ V
31, 2dfop 3589 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 3660 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 3977 . . . . . 6 {𝐴} ∈ V
6 prexg 3994 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 417 . . . . . 6 {𝐴, 𝐵} ∈ V
85, 7intpr 3688 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9 snsspr1 3553 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
10 df-ss 2995 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
119, 10mpbi 143 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
128, 11eqtri 2103 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
134, 12eqtri 2103 . . 3 𝐴, 𝐵⟩ = {𝐴}
1413inteqi 3660 . 2 𝐴, 𝐵⟩ = {𝐴}
151intsn 3691 . 2 {𝐴} = 𝐴
1614, 15eqtri 2103 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  Vcvv 2610  cin 2981  wss 2982  {csn 3416  {cpr 3417  cop 3419   cint 3656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-int 3657
This theorem is referenced by:  elreldm  4608  op2ndb  4854  1stval2  5834  fundmen  6375  xpsnen  6387
  Copyright terms: Public domain W3C validator