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Theorem elxp7 8010
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7913. (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 8009 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 fvex 6905 . . . . 5 (1st𝐴) ∈ V
3 fvex 6905 . . . . 5 (2nd𝐴) ∈ V
42, 3pm3.2i 472 . . . 4 ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)
5 elxp6 8009 . . . 4 (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)))
64, 5mpbiran2 709 . . 3 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76anbi1i 625 . 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 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cop 4635   × cxp 5675  cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  xp2  8012  unielxp  8013  1stconst  8086  2ndconst  8087  fparlem1  8098  fparlem2  8099  infxpenlem  10008  1stpreimas  31927  1stpreima  31928  2ndpreima  31929  f1od2  31946  xpinpreima2  32887  tpr2rico  32892  sxbrsigalem0  33270  dya2iocnrect  33280  elxp8  36252  pellex  41573  elpglem3  47758
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