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Theorem op1stbg 4479
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 3776 . . . . 5 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21inteqd 3849 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 snexg 4184 . . . . . 6 (𝐴𝑉 → {𝐴} ∈ V)
4 prexg 4211 . . . . . 6 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
5 intprg 3877 . . . . . 6 (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
63, 4, 5syl2an2r 595 . . . . 5 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
7 snsspr1 3740 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
8 df-ss 3142 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
97, 8mpbi 145 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
106, 9eqtrdi 2226 . . . 4 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = {𝐴})
112, 10eqtrd 2210 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
1211inteqd 3849 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
13 intsng 3878 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
1413adantr 276 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} = 𝐴)
1512, 14eqtrd 2210 1 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cin 3128  wss 3129  {csn 3592  {cpr 3593  cop 3595   cint 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-int 3845
This theorem is referenced by:  elxp5  5117  fundmen  6805
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