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Theorem op1stb 4209
 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 3548 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 3619 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snexgOLD 3935 . . . . . . 7 (𝐴 ∈ V → {𝐴} ∈ V)
61, 5ax-mp 7 . . . . . 6 {𝐴} ∈ V
7 prexgOLD 3946 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
81, 2, 7mp2an 402 . . . . . 6 {𝐴, 𝐵} ∈ V
96, 8intpr 3647 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
10 snsspr1 3512 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
11 df-ss 2931 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
1210, 11mpbi 133 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
139, 12eqtri 2060 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
144, 13eqtri 2060 . . 3 𝐴, 𝐵⟩ = {𝐴}
1514inteqi 3619 . 2 𝐴, 𝐵⟩ = {𝐴}
161intsn 3650 . 2 {𝐴} = 𝐴
1715, 16eqtri 2060 1 𝐴, 𝐵⟩ = 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917  {csn 3375  {cpr 3376  ⟨cop 3378  ∩ cint 3615 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-int 3616 This theorem is referenced by:  elreldm  4560  op2ndb  4804  1stval2  5782  fundmen  6286  xpsnen  6295
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