Theorem opelxp 4811

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>. e. (C X. D) <-> (A e. C /\ B e. 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 4602	. . . . . 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 e. C /\ y e. D)) <-> ((x = A /\ y = B) /\ (x e. C /\ y e. D)))
5		an4 797	. . . 4 |- (((x = A /\ y = B) /\ (x e. C /\ y e. D)) <-> ((x = A /\ x e. C) /\ (y = B /\ y e. D)))
6	4, 5	bitri 240	. . 3 |- ((<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> ((x = A /\ x e. C) /\ (y = B /\ y e. D)))
7	6	2exbii 1583	. 2 |- (E.xE.y(<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> E.xE.y((x = A /\ x e. C) /\ (y = B /\ y e. D)))
8		elxp 4801	. 2 |- (<.A, B>. e. (C X. D) <-> E.xE.y(<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)))
9		df-clel 2349	. . . 4 |- (A e. C <-> E.x(x = A /\ x e. C))
10		df-clel 2349	. . . 4 |- (B e. D <-> E.y(y = B /\ y e. D))
11	9, 10	anbi12i 678	. . 3 |- ((A e. C /\ B e. D) <-> (E.x(x = A /\ x e. C) /\ E.y(y = B /\ y e. D)))
12		eeanv 1913	. . 3 |- (E.xE.y((x = A /\ x e. C) /\ (y = B /\ y e. D)) <-> (E.x(x = A /\ x e. C) /\ E.y(y = B /\ y e. D)))
13	11, 12	bitr4i 243	. 2 |- ((A e. C /\ B e. D) <-> E.xE.y((x = A /\ x e. C) /\ (y = B /\ y e. D)))
14	7, 8, 13	3bitr4i 268	1 |- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  <.cop 4561   X. 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-13 1712  ax-14 1714  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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-xp 4784

This theorem is referenced by:  brxp  4812  opeliunxp  4820  elxp3  4831  optocl  4838  opbrop  4841  xpvv  4843  xpnz  5045  ssrnres  5059  dfco2  5080  ssdmrn  5099  opelf  5235  ressnop0  5436  xpnedisj  5513  fnopovb  5557  oprab4  5566  resoprab  5581  ov3  5599  ovg  5601  ovres  5602  fovrn  5604  fnovrn  5607  ovconst2  5611  oprssdm  5612  ndmovg  5613  ndmovcl  5614  ndmov  5615  releqmpt  5808  releqmpt2  5809  composeex  5820  fnsex  5832  crossex  5850  transex  5910  foundex  5914  ecopqsi  5981  xpassen  6057  enprmaplem4  6079  ovcelem1  6171  tcfnex  6244  csucex  6259  nmembers1lem1  6268  nncdiv3lem2  6276  spacvallem1  6281  nchoicelem11  6299  frecxp  6314  fnfreclem1  6317