Theorem elxp2 4802

Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)

Assertion

Ref	Expression
elxp2	⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = ⟨x, y⟩)

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

Proof of Theorem elxp2

Step	Hyp	Ref	Expression
1		df-rex 2620	. . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = ⟨x, y⟩) ↔ ∃y(y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)))
2		r19.42v 2765	. . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = ⟨x, y⟩) ↔ (x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩))
3		an13 774	. . . . 5 ⊢ ((y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)) ↔ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
4	3	exbii 1582	. . . 4 ⊢ (∃y(y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)) ↔ ∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
5	1, 2, 4	3bitr3i 266	. . 3 ⊢ ((x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩) ↔ ∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
6	5	exbii 1582	. 2 ⊢ (∃x(x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
7		df-rex 2620	. 2 ⊢ (∃x ∈ B ∃y ∈ C A = ⟨x, y⟩ ↔ ∃x(x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩))
8		elxp 4801	. 2 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
9	6, 7, 8	3bitr4ri 269	1 ⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = ⟨x, y⟩)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟨cop 4561   × 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:  xpiundi  4817  xpiundir  4818  dfxp2  5113  xpnedisj  5513  1st2nd2  5516  crossex  5850  xpassen  6057  addccan2nclem1  6263