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Theorem op1stb 4901
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5578 to extract the second member, op1sta 5576 for an alternate version, and op1st 7121 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 4369 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 4444 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 4869 . . . . . 6 {𝐴} ∈ V
6 prex 4870 . . . . . 6 {𝐴, 𝐵} ∈ V
75, 6intpr 4475 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8 snsspr1 4313 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 df-ss 3569 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
108, 9mpbi 220 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
117, 10eqtri 2643 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
124, 11eqtri 2643 . . 3 𝐴, 𝐵⟩ = {𝐴}
1312inteqi 4444 . 2 𝐴, 𝐵⟩ = {𝐴}
141intsn 4478 . 2 {𝐴} = 𝐴
1513, 14eqtri 2643 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  {csn 4148  {cpr 4150  cop 4154   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-int 4441
This theorem is referenced by:  elreldm  5310  op2ndb  5578  elxp5  7058  1stval2  7130  fundmen  7974  xpsnen  7988
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