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Theorem op1stb 4273
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 3604 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 3675 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 3994 . . . . . 6 {𝐴} ∈ V
6 prexg 4012 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 417 . . . . . 6 {𝐴, 𝐵} ∈ V
85, 7intpr 3703 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9 snsspr1 3568 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
10 df-ss 3001 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
119, 10mpbi 143 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
128, 11eqtri 2105 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
134, 12eqtri 2105 . . 3 𝐴, 𝐵⟩ = {𝐴}
1413inteqi 3675 . 2 𝐴, 𝐵⟩ = {𝐴}
151intsn 3706 . 2 {𝐴} = 𝐴
1614, 15eqtri 2105 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  Vcvv 2615  cin 2987  wss 2988  {csn 3431  {cpr 3432  cop 3434   cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-int 3672
This theorem is referenced by:  elreldm  4629  op2ndb  4880  1stval2  5883  fundmen  6475  xpsnen  6489
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