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Theorem elxp2 4306
 Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
Assertion
Ref Expression
elxp2 (A (B × 𝐶) ↔ x B y 𝐶 A = ⟨x, y⟩)
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem elxp2
StepHypRef Expression
1 df-rex 2306 . . . 4 (y 𝐶 (x B A = ⟨x, y⟩) ↔ y(y 𝐶 (x B A = ⟨x, y⟩)))
2 r19.42v 2461 . . . 4 (y 𝐶 (x B A = ⟨x, y⟩) ↔ (x B y 𝐶 A = ⟨x, y⟩))
3 an13 497 . . . . 5 ((y 𝐶 (x B A = ⟨x, y⟩)) ↔ (A = ⟨x, y (x B y 𝐶)))
43exbii 1493 . . . 4 (y(y 𝐶 (x B A = ⟨x, y⟩)) ↔ y(A = ⟨x, y (x B y 𝐶)))
51, 2, 43bitr3i 199 . . 3 ((x B y 𝐶 A = ⟨x, y⟩) ↔ y(A = ⟨x, y (x B y 𝐶)))
65exbii 1493 . 2 (x(x B y 𝐶 A = ⟨x, y⟩) ↔ xy(A = ⟨x, y (x B y 𝐶)))
7 df-rex 2306 . 2 (x B y 𝐶 A = ⟨x, y⟩ ↔ x(x B y 𝐶 A = ⟨x, y⟩))
8 elxp 4305 . 2 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
96, 7, 83bitr4ri 202 1 (A (B × 𝐶) ↔ x B y 𝐶 A = ⟨x, y⟩)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370   × cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294 This theorem is referenced by:  opelxp  4317  xpiundi  4341  xpiundir  4342  ssrel2  4373  f1o2ndf1  5791  xpdom2  6241  elreal  6707
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