Theorem opkeq1 4060

Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)

Assertion

Ref	Expression
opkeq1	|- (A = B -> <<A, C>> = <<B, C>>)

Proof of Theorem opkeq1

Step	Hyp	Ref	Expression
1		sneq 3745	. . 3 |- (A = B -> {A} = {B})
2		preq1 3800	. . 3 |- (A = B -> {A, C} = {B, C})
3	1, 2	preq12d 3808	. 2 |- (A = B -> {{A}, {A, C}} = {{B}, {B, C}})
4		df-opk 4059	. 2 |- <<A, C>> = {{A}, {A, C}}
5		df-opk 4059	. 2 |- <<B, C>> = {{B}, {B, C}}
6	3, 4, 5	3eqtr4g 2410	1 |- (A = B -> <<A, C>> = <<B, C>>)

Syntax hints:   -> wi 4   = wceq 1642  {csn 3738  {cpr 3739  <<copk 4058

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  opkeq12  4062  opkeq1i  4063  opkeq1d  4066  opkthg  4132  opkelcnvkg  4250  otkelins2kg  4254  otkelins3kg  4255  opkelcokg  4262  opksnelsik  4266  opkelimagekg  4272  elimaksn  4284  sikexlem  4296  dfimak2  4299  insklem  4305  setswith  4322  ndisjrelk  4324  dfpw2  4328  dfaddc2  4382  dfnnc2  4396  nnsucelrlem1  4425  leltfintr  4459  ltfinex  4465  ltfintrilem1  4466  ltfintri  4467  ssfin  4471  eqpwrelk  4479  eqpw1relk  4480  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelk  4525  sfintfinlem1  4532  tfinnnlem1  4534  sfinltfin  4536  spfinex  4538  vfinncvntnn  4549  vfinspss  4552  vfinncsp  4555  dfop2lem1  4574  setconslem1  4732  setconslem2  4733  setconslem3  4734  setconslem4  4735  setconslem6  4737  setconslem7  4738  df1st2  4739  dfswap2  4742