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Theorem elxpi 4304
Description: Membership in a cross product. Uses fewer axioms than elxp 4305. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxpi  X.  C  <. ,  >.  C
Distinct variable groups:   ,,   ,,   , C,

Proof of Theorem elxpi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . 6  <. ,  >.  <. ,  >.
21anbi1d 438 . . . . 5  <. ,  >.  C  <. ,  >.  C
322exbidv 1745 . . . 4  <. ,  >.  C  <. ,  >.  C
43elabg 2682 . . 3  {  |  <. ,  >.  C }  {  |  <. ,  >.  C }  <. ,  >.  C
54ibi 165 . 2  {  |  <. ,  >.  C }  <. ,  >.  C
6 df-xp 4294 . . 3  X.  C  { <. , 
>.  |  C }
7 df-opab 3810 . . 3  { <. ,  >.  |  C }  {  |  <. ,  >.  C }
86, 7eqtri 2057 . 2  X.  C  {  |  <. , 
>.  C }
95, 8eleq2s 2129 1  X.  C  <. ,  >.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023   <.cop 3370   {copab 3808    X. 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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpsspw  4393  dmaddpqlem  6361  nqpi  6362  enq0ref  6416  nqnq0  6424  nq0nn  6425  axaddcl  6750  axmulcl  6752
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