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Theorem op1stb 5354
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6077 to extract the second member, op1sta 6075 for an alternate version, and op1st 7686 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 4794 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 4871 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5322 . . . . . 6 {𝐴} ∈ V
6 prex 5323 . . . . . 6 {𝐴, 𝐵} ∈ V
75, 6intpr 4900 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8 snsspr1 4739 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 df-ss 3949 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
108, 9mpbi 231 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
117, 10eqtri 2841 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
124, 11eqtri 2841 . . 3 𝐴, 𝐵⟩ = {𝐴}
1312inteqi 4871 . 2 𝐴, 𝐵⟩ = {𝐴}
141intsn 4903 . 2 {𝐴} = 𝐴
1513, 14eqtri 2841 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  wss 3933  {csn 4557  {cpr 4559  cop 4563   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-int 4868
This theorem is referenced by:  elreldm  5798  op2ndb  6077  elxp5  7617  1stval2  7695  fundmen  8571  xpsnen  8589
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