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Theorem op1stbg 4291
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
Assertion
Ref Expression
op1stbg ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)

Proof of Theorem op1stbg
StepHypRef Expression
1 dfopg 3615 . . . . 5 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21inteqd 3688 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 snexg 4010 . . . . . 6 (𝐴𝑉 → {𝐴} ∈ V)
4 prexg 4029 . . . . . 6 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
5 intprg 3716 . . . . . 6 (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
63, 4, 5syl2an2r 562 . . . . 5 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
7 snsspr1 3580 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
8 df-ss 3010 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
97, 8mpbi 143 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
106, 9syl6eq 2136 . . . 4 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = {𝐴})
112, 10eqtrd 2120 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
1211inteqd 3688 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
13 intsng 3717 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
1413adantr 270 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} = 𝐴)
1512, 14eqtrd 2120 1 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  cin 2996  wss 2997  {csn 3441  {cpr 3442  cop 3444   cint 3683
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684
This theorem is referenced by:  elxp5  4906  fundmen  6503
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