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Theorem op1stb 5476
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6247 to extract the second member, op1sta 6245 for an alternate version, and op1st 8022 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 4872 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 4950 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5436 . . . . . 6 {𝐴} ∈ V
6 prex 5437 . . . . . 6 {𝐴, 𝐵} ∈ V
75, 6intpr 4982 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8 snsspr1 4814 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 dfss2 3969 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
108, 9mpbi 230 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
117, 10eqtri 2765 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
124, 11eqtri 2765 . . 3 𝐴, 𝐵⟩ = {𝐴}
1312inteqi 4950 . 2 𝐴, 𝐵⟩ = {𝐴}
141intsn 4984 . 2 {𝐴} = 𝐴
1513, 14eqtri 2765 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  {csn 4626  {cpr 4628  cop 4632   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-int 4947
This theorem is referenced by:  elreldm  5946  op2ndb  6247  elxp5  7945  1stval2  8031  fundmen  9071  xpsnen  9095
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