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Theorem op1stb 4359
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 3670 . . . . 5 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43inteqi 3741 . . . 4 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 4069 . . . . . 6 {𝐴} ∈ V
6 prexg 4093 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 420 . . . . . 6 {𝐴, 𝐵} ∈ V
85, 7intpr 3769 . . . . 5 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9 snsspr1 3634 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
10 df-ss 3050 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
119, 10mpbi 144 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
128, 11eqtri 2135 . . . 4 {{𝐴}, {𝐴, 𝐵}} = {𝐴}
134, 12eqtri 2135 . . 3 𝐴, 𝐵⟩ = {𝐴}
1413inteqi 3741 . 2 𝐴, 𝐵⟩ = {𝐴}
151intsn 3772 . 2 {𝐴} = 𝐴
1614, 15eqtri 2135 1 𝐴, 𝐵⟩ = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wcel 1463  Vcvv 2657  cin 3036  wss 3037  {csn 3493  {cpr 3494  cop 3496   cint 3737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-int 3738
This theorem is referenced by:  elreldm  4725  op2ndb  4980  1stval2  6007  fundmen  6654  xpsnen  6668
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