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Theorem op1stb 5472
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6227 to extract the second member, op1sta 6225 for an alternate version, and op1st 7983 for the preferred version.) (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 4873 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 4955 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5432 . . . . . 6 {𝐴} ∈ V
6 prex 5433 . . . . . 6 {𝐴, 𝐵} ∈ V
75, 6intpr 4987 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8 snsspr1 4818 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 df-ss 3966 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
108, 9mpbi 229 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
117, 10eqtri 2761 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
124, 11eqtri 2761 . . 3 𝐴, 𝐵⟩ = {𝐴}
1312inteqi 4955 . 2 𝐴, 𝐵⟩ = {𝐴}
141intsn 4991 . 2 {𝐴} = 𝐴
1513, 14eqtri 2761 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  wss 3949  {csn 4629  {cpr 4631  cop 4635   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-int 4952
This theorem is referenced by:  elreldm  5935  op2ndb  6227  elxp5  7914  1stval2  7992  fundmen  9031  xpsnen  9055
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