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Theorem elxpi 4284
Description: Membership in a cross product. Uses fewer axioms than elxp 4285. (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 2024 . . . . . 6 (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
21anbi1d 441 . . . . 5 (z = A → ((z = ⟨x, y (x B y 𝐶)) ↔ (A = ⟨x, y (x B y 𝐶))))
322exbidv 1726 . . . 4 (z = A → (xy(z = ⟨x, y (x B y 𝐶)) ↔ xy(A = ⟨x, y (x B y 𝐶))))
43elabg 2661 . . 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 4274 . . 3 (B × 𝐶) = {⟨x, y⟩ ∣ (x B y 𝐶)}
7 df-opab 3789 . . 3 {⟨x, y⟩ ∣ (x B y 𝐶)} = {zxy(z = ⟨x, y (x B y 𝐶))}
86, 7eqtri 2038 . 2 (B × 𝐶) = {zxy(z = ⟨x, y (x B y 𝐶))}
95, 8eleq2s 2110 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 1226  wex 1358   wcel 1370  {cab 2004  cop 3349  {copab 3787   × 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-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-opab 3789  df-xp 4274
This theorem is referenced by:  xpsspw  4373  dmaddpqlem  6230  nqpi  6231  enq0ref  6282  nqnq0  6290  nq0nn  6291  axaddcl  6554  axmulcl  6556
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