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Theorem op1stbg 4464
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 3763 . . . . 5 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21inteqd 3836 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 snexg 4170 . . . . . 6 (𝐴𝑉 → {𝐴} ∈ V)
4 prexg 4196 . . . . . 6 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
5 intprg 3864 . . . . . 6 (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
63, 4, 5syl2an2r 590 . . . . 5 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
7 snsspr1 3728 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
8 df-ss 3134 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
97, 8mpbi 144 . . . . 5 ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
106, 9eqtrdi 2219 . . . 4 ((𝐴𝑉𝐵𝑊) → {{𝐴}, {𝐴, 𝐵}} = {𝐴})
112, 10eqtrd 2203 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
1211inteqd 3836 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = {𝐴})
13 intsng 3865 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
1413adantr 274 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} = 𝐴)
1512, 14eqtrd 2203 1 ((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  cin 3120  wss 3121  {csn 3583  {cpr 3584  cop 3586   cint 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-int 3832
This theorem is referenced by:  elxp5  5099  fundmen  6784
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