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Theorem elxp7 7957
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7860. (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 7956 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
2 fvex 6856 . . . . 5 (1st𝐴) ∈ V
3 fvex 6856 . . . . 5 (2nd𝐴) ∈ V
42, 3pm3.2i 472 . . . 4 ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)
5 elxp6 7956 . . . 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 3446  cop 4593   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  xp2  7959  unielxp  7960  1stconst  8033  2ndconst  8034  fparlem1  8045  fparlem2  8046  infxpenlem  9950  1stpreimas  31622  1stpreima  31623  2ndpreima  31624  f1od2  31641  xpinpreima2  32491  tpr2rico  32496  sxbrsigalem0  32874  dya2iocnrect  32884  elxp8  35845  pellex  41161  elpglem3  47165
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