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Theorem op1stbg 4238
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 3575 . . . . 5 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21inteqd 3648 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 elex 2583 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
4 snexgOLD 3963 . . . . . . . 8 (𝐴 ∈ V → {𝐴} ∈ V)
53, 4syl 14 . . . . . . 7 (𝐴𝑉 → {𝐴} ∈ V)
65adantr 265 . . . . . 6 ((𝐴𝑉𝐵𝑊) → {𝐴} ∈ V)
7 elex 2583 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
8 prexgOLD 3974 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
93, 7, 8syl2an 277 . . . . . 6 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
10 intprg 3676 . . . . . 6 (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
116, 9, 10syl2anc 397 . . . . 5 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
12 snsspr1 3540 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
13 df-ss 2959 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
1412, 13mpbi 137 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
1511, 14syl6eq 2104 . . . 4 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = {𝐴})
162, 15eqtrd 2088 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
1716inteqd 3648 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
18 intsng 3677 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
1918adantr 265 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} = 𝐴)
2017, 19eqtrd 2088 1 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cin 2944  wss 2945  {csn 3403  {cpr 3404  cop 3406   cint 3643
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 3903  ax-pow 3955  ax-pr 3972
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 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-int 3644
This theorem is referenced by:  elxp5  4837  fundmen  6317
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