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Theorem op1stb 4175
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (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 3539 . . . . 5 A, B⟩ = {{A}, {A, B}}
43inteqi 3610 . . . 4 A, B⟩ = {{A}, {A, B}}
5 snexgOLD 3926 . . . . . . 7 (A V → {A} V)
61, 5ax-mp 7 . . . . . 6 {A} V
7 prexgOLD 3937 . . . . . . 7 ((A V B V) → {A, B} V)
81, 2, 7mp2an 402 . . . . . 6 {A, B} V
96, 8intpr 3638 . . . . 5 {{A}, {A, B}} = ({A} ∩ {A, B})
10 snsspr1 3503 . . . . . 6 {A} ⊆ {A, B}
11 df-ss 2925 . . . . . 6 ({A} ⊆ {A, B} ↔ ({A} ∩ {A, B}) = {A})
1210, 11mpbi 133 . . . . 5 ({A} ∩ {A, B}) = {A}
139, 12eqtri 2057 . . . 4 {{A}, {A, B}} = {A}
144, 13eqtri 2057 . . 3 A, B⟩ = {A}
1514inteqi 3610 . 2 A, B⟩ = {A}
161intsn 3641 . 2 {A} = A
1715, 16eqtri 2057 1 A, B⟩ = A
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  wss 2911  {csn 3367  {cpr 3368  cop 3370   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607
This theorem is referenced by:  elreldm  4503  op2ndb  4747  1stval2  5724  fundmen  6222  xpsnen  6231
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