Theorem opelxp 4812

Description: Ordered pair membership in a cross product. (The proof was shortened by Andrew Salmon, 12-Aug-2011.) (Contributed by NM, 15-Nov-1994.) (Revised by set.mm contributors, 12-Aug-2011.)

Assertion

Ref	Expression
opelxp	⊢ (⟨A, B⟩ ∈ (C × D) ↔ (A ∈ C ∧ B ∈ D))

Proof of Theorem opelxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2355	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨A, B⟩)
2		opth 4603	. . . . . 6 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
3	1, 2	bitri 240	. . . . 5 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ (x = A ∧ y = B))
4	3	anbi1i 676	. . . 4 ⊢ ((⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x = A ∧ y = B) ∧ (x ∈ C ∧ y ∈ D)))
5		an4 797	. . . 4 ⊢ (((x = A ∧ y = B) ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x = A ∧ x ∈ C) ∧ (y = B ∧ y ∈ D)))
6	4, 5	bitri 240	. . 3 ⊢ ((⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x = A ∧ x ∈ C) ∧ (y = B ∧ y ∈ D)))
7	6	2exbii 1583	. 2 ⊢ (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)) ↔ ∃x∃y((x = A ∧ x ∈ C) ∧ (y = B ∧ y ∈ D)))
8		elxp 4802	. 2 ⊢ (⟨A, B⟩ ∈ (C × D) ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)))
9		df-clel 2349	. . . 4 ⊢ (A ∈ C ↔ ∃x(x = A ∧ x ∈ C))
10		df-clel 2349	. . . 4 ⊢ (B ∈ D ↔ ∃y(y = B ∧ y ∈ D))
11	9, 10	anbi12i 678	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) ↔ (∃x(x = A ∧ x ∈ C) ∧ ∃y(y = B ∧ y ∈ D)))
12		eeanv 1913	. . 3 ⊢ (∃x∃y((x = A ∧ x ∈ C) ∧ (y = B ∧ y ∈ D)) ↔ (∃x(x = A ∧ x ∈ C) ∧ ∃y(y = B ∧ y ∈ D)))
13	11, 12	bitr4i 243	. 2 ⊢ ((A ∈ C ∧ B ∈ D) ↔ ∃x∃y((x = A ∧ x ∈ C) ∧ (y = B ∧ y ∈ D)))
14	7, 8, 13	3bitr4i 268	1 ⊢ (⟨A, B⟩ ∈ (C × D) ↔ (A ∈ C ∧ B ∈ D))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4562   × cxp 4771

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785

This theorem is referenced by:  brxp  4813  opeliunxp  4821  elxp3  4832  optocl  4839  opbrop  4842  xpvv  4844  xpnz  5046  ssrnres  5060  dfco2  5081  ssdmrn  5100  opelf  5236  ressnop0  5437  xpnedisj  5514  fnopovb  5558  oprab4  5567  resoprab  5582  ov3  5600  ovg  5602  ovres  5603  fovrn  5605  fnovrn  5608  ovconst2  5612  oprssdm  5613  ndmovg  5614  ndmovcl  5615  ndmov  5616  releqmpt  5809  releqmpt2  5810  composeex  5821  fnsex  5833  crossex  5851  transex  5911  foundex  5915  ecopqsi  5982  xpassen  6058  enprmaplem4  6080  ovcelem1  6172  tcfnex  6245  csucex  6260  nmembers1lem1  6269  nncdiv3lem2  6277  spacvallem1  6282  nchoicelem11  6300  frecxp  6315  fnfreclem1  6318