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Theorem elxp7 5824
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (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 elex 2583 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 elex 2583 . . 3 (𝐴 ∈ (V × V) → 𝐴 ∈ V)
32adantr 265 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
4 1stexg 5821 . . . . . . 7 (𝐴 ∈ V → (1st𝐴) ∈ V)
5 2ndexg 5822 . . . . . . 7 (𝐴 ∈ V → (2nd𝐴) ∈ V)
64, 5jca 294 . . . . . 6 (𝐴 ∈ V → ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V))
76biantrud 292 . . . . 5 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V))))
8 elxp6 5823 . . . . 5 (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)))
97, 8syl6rbbr 192 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩))
109anbi1d 446 . . 3 (𝐴 ∈ V → ((𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
11 elxp6 5823 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1210, 11syl6rbbr 192 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
131, 3, 12pm5.21nii 630 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  cop 3405   × cxp 4370  cfv 4929  1st c1st 5792  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-1st 5794  df-2nd 5795
This theorem is referenced by:  xp2  5826  unielxp  5827  1stconst  5869  2ndconst  5870  f1od2  5883
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