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Theorem elxp3 4337
Description: Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
Assertion
Ref Expression
elxp3 (A (B × 𝐶) ↔ xy(⟨x, y⟩ = A x, y (B × 𝐶)))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem elxp3
StepHypRef Expression
1 elxp 4305 . 2 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
2 eqcom 2039 . . . 4 (⟨x, y⟩ = AA = ⟨x, y⟩)
3 opelxp 4317 . . . 4 (⟨x, y (B × 𝐶) ↔ (x B y 𝐶))
42, 3anbi12i 433 . . 3 ((⟨x, y⟩ = A x, y (B × 𝐶)) ↔ (A = ⟨x, y (x B y 𝐶)))
542exbii 1494 . 2 (xy(⟨x, y⟩ = A x, y (B × 𝐶)) ↔ xy(A = ⟨x, y (x B y 𝐶)))
61, 5bitr4i 176 1 (A (B × 𝐶) ↔ xy(⟨x, y⟩ = A x, y (B × 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  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-ral 2305  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:  optocl  4359
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