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Theorem elxp7 8049
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7944. (Contributed by NM, 19-Aug-2006.)
Assertion
Ref Expression
elxp7 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Proof of Theorem elxp7
StepHypRef Expression
1 elxp6 8048 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 fvex 6919 . . . . 5 (1st𝐴) ∈ V
3 fvex 6919 . . . . 5 (2nd𝐴) ∈ V
42, 3pm3.2i 470 . . . 4 ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)
5 elxp6 8048 . . . 4 (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)))
64, 5mpbiran2 710 . . 3 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76anbi1i 624 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
81, 7bitr4i 278 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  xp2  8051  unielxp  8052  1stconst  8125  2ndconst  8126  fparlem1  8137  fparlem2  8138  infxpenlem  10053  1stpreimas  32715  1stpreima  32716  2ndpreima  32717  f1od2  32732  xpinpreima2  33906  tpr2rico  33911  sxbrsigalem0  34273  dya2iocnrect  34283  elxp8  37372  pellex  42846  elpglem3  49232
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