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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 (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 elxpi
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . 6 (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
21anbi1d 438 . . . . 5 (z = A → ((z = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, y (x B y 𝐶))))
322exbidv 1745 . . . 4 (z = A → (xy(z = ⟨x, y (x B y 𝐶)) ↔ xy(A = ⟨x, y (x B y 𝐶))))
43elabg 2682 . . 3 (A {zxy(z = ⟨x, y (x B y 𝐶))} → (A {zxy(z = ⟨x, y (x B y 𝐶))} ↔ xy(A = ⟨x, y (x B y 𝐶))))
54ibi 165 . 2 (A {zxy(z = ⟨x, y (x B y 𝐶))} → xy(A = ⟨x, y (x B y 𝐶)))
6 df-xp 4294 . . 3 (B × 𝐶) = {⟨x, y⟩ ∣ (x B y 𝐶)}
7 df-opab 3810 . . 3 {⟨x, y⟩ ∣ (x B y 𝐶)} = {zxy(z = ⟨x, y (x B y 𝐶))}
86, 7eqtri 2057 . 2 (B × 𝐶) = {zxy(z = ⟨x, y (x B y 𝐶))}
95, 8eleq2s 2129 1 (A (B × 𝐶) → xy(A = ⟨x, y (x B y 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   × 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-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  6415  nqnq0  6423  nq0nn  6424  axaddcl  6730  axmulcl  6732
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