Theorem elxp 4801

Description: Membership in a cross product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
elxp	⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))

Distinct variable groups:   x,y,A   x,B,y   x,C,y

Proof of Theorem elxp

Step	Hyp	Ref	Expression
1		df-xp 4784	. . 3 ⊢ (B × C) = {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ (B × C) ↔ A ∈ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)})
3		elopab 4696	. 2 ⊢ (A ∈ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)} ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
4	2, 3	bitri 240	1 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622   × cxp 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-xp 4784

This theorem is referenced by:  elxp2  4802  opelxp  4811  rabxp  4814  elxp3  4831  xp0r  4842  dfco2a  5081  elxp4  5108  fnov  5591