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Theorem elxp3 4317
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 4285 . 2 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
2 eqcom 2020 . . . 4 (⟨x, y⟩ = AA = ⟨x, y⟩)
3 opelxp 4297 . . . 4 (⟨x, y (B × 𝐶) ↔ (x B y 𝐶))
42, 3anbi12i 436 . . 3 ((⟨x, y⟩ = A x, y (B × 𝐶)) ↔ (A = ⟨x, y (x B y 𝐶)))
542exbii 1475 . 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 1226  wex 1358   wcel 1370  cop 3349   × cxp 4266
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274
This theorem is referenced by:  optocl  4339
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