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Theorem op1stb 5350
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6071 to extract the second member, op1sta 6069 for an alternate version, and op1st 7692 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 4786 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 4866 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5319 . . . . . 6 {𝐴} ∈ V
6 prex 5320 . . . . . 6 {𝐴, 𝐵} ∈ V
75, 6intpr 4895 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8 snsspr1 4731 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 df-ss 3936 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
108, 9mpbi 233 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
117, 10eqtri 2847 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
124, 11eqtri 2847 . . 3 𝐴, 𝐵⟩ = {𝐴}
1312inteqi 4866 . 2 𝐴, 𝐵⟩ = {𝐴}
141intsn 4898 . 2 {𝐴} = 𝐴
1513, 14eqtri 2847 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918  wss 3919  {csn 4550  {cpr 4552  cop 4556   cint 4862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-int 4863
This theorem is referenced by:  elreldm  5792  op2ndb  6071  elxp5  7623  1stval2  7701  fundmen  8579  xpsnen  8597
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