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Theorem op1stbg 4176
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 ((A 𝑉 B 𝑊) → A, B⟩ = A)

Proof of Theorem op1stbg
StepHypRef Expression
1 dfopg 3538 . . . . 5 ((A 𝑉 B 𝑊) → ⟨A, B⟩ = {{A}, {A, B}})
21inteqd 3611 . . . 4 ((A 𝑉 B 𝑊) → A, B⟩ = {{A}, {A, B}})
3 elex 2560 . . . . . . . 8 (A 𝑉A V)
4 snexgOLD 3926 . . . . . . . 8 (A V → {A} V)
53, 4syl 14 . . . . . . 7 (A 𝑉 → {A} V)
65adantr 261 . . . . . 6 ((A 𝑉 B 𝑊) → {A} V)
7 elex 2560 . . . . . . 7 (B 𝑊B V)
8 prexgOLD 3937 . . . . . . 7 ((A V B V) → {A, B} V)
93, 7, 8syl2an 273 . . . . . 6 ((A 𝑉 B 𝑊) → {A, B} V)
10 intprg 3639 . . . . . 6 (({A} V {A, B} V) → {{A}, {A, B}} = ({A} ∩ {A, B}))
116, 9, 10syl2anc 391 . . . . 5 ((A 𝑉 B 𝑊) → {{A}, {A, B}} = ({A} ∩ {A, B}))
12 snsspr1 3503 . . . . . 6 {A} ⊆ {A, B}
13 df-ss 2925 . . . . . 6 ({A} ⊆ {A, B} ↔ ({A} ∩ {A, B}) = {A})
1412, 13mpbi 133 . . . . 5 ({A} ∩ {A, B}) = {A}
1511, 14syl6eq 2085 . . . 4 ((A 𝑉 B 𝑊) → {{A}, {A, B}} = {A})
162, 15eqtrd 2069 . . 3 ((A 𝑉 B 𝑊) → A, B⟩ = {A})
1716inteqd 3611 . 2 ((A 𝑉 B 𝑊) → A, B⟩ = {A})
18 intsng 3640 . . 3 (A 𝑉 {A} = A)
1918adantr 261 . 2 ((A 𝑉 B 𝑊) → {A} = A)
2017, 19eqtrd 2069 1 ((A 𝑉 B 𝑊) → A, B⟩ = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  wss 2911  {csn 3367  {cpr 3368  cop 3370   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607
This theorem is referenced by:  elxp5  4752  fundmen  6222
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