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Theorem elxp 4289
Description: Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxp (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem elxp
StepHypRef Expression
1 df-xp 4278 . . 3 (B × 𝐶) = {⟨x, y⟩ ∣ (x B y 𝐶)}
21eleq2i 2086 . 2 (A (B × 𝐶) ↔ A {⟨x, y⟩ ∣ (x B y 𝐶)})
3 elopab 3969 . 2 (A {⟨x, y⟩ ∣ (x B y 𝐶)} ↔ xy(A = ⟨x, y (x B y 𝐶)))
42, 3bitri 173 1 (A (B × 𝐶) ↔ xy(A = ⟨x, y (x B y 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  cop 3353  {copab 3791   × cxp 4270
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793  df-xp 4278
This theorem is referenced by:  elxp2  4290  0nelxp  4299  0nelelxp  4300  rabxp  4307  elxp3  4321  elvv  4329  elvvv  4330  0xp  4347  xpmlem  4671  elxp4  4735  elxp5  4736  dfco2a  4748  opabex3d  5671  opabex3  5672  xp1st  5715  xp2nd  5716  poxp  5775  nqnq0pi  6293
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