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Theorem op1stb 4236
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 3575 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 3646 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snexgOLD 3962 . . . . . . 7 (𝐴 ∈ V → {𝐴} ∈ V)
61, 5ax-mp 7 . . . . . 6 {𝐴} ∈ V
7 prexgOLD 3973 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
81, 2, 7mp2an 410 . . . . . 6 {𝐴, 𝐵} ∈ V
96, 8intpr 3674 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
10 snsspr1 3539 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
11 df-ss 2958 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
1210, 11mpbi 137 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
139, 12eqtri 2076 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
144, 13eqtri 2076 . . 3 𝐴, 𝐵⟩ = {𝐴}
1514inteqi 3646 . 2 𝐴, 𝐵⟩ = {𝐴}
161intsn 3677 . 2 {𝐴} = 𝐴
1715, 16eqtri 2076 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cin 2943  wss 2944  {csn 3402  {cpr 3403  cop 3405   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-int 3643
This theorem is referenced by:  elreldm  4587  op2ndb  4831  1stval2  5809  fundmen  6316  xpsnen  6325
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