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Theorem opklefing 4448
 Description: Kuratowski ordered pair membership in finite less than or equal to. (Contributed by SF, 18-Jan-2015.)
Assertion
Ref Expression
opklefing fin Nn
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem opklefing
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lefin 4440 . 2 fin Nn
2 addceq1 4383 . . . 4
32eqeq2d 2364 . . 3
43rexbidv 2635 . 2 Nn Nn
5 eqeq1 2359 . . 3
65rexbidv 2635 . 2 Nn Nn
71, 4, 6opkelopkabg 4245 1 fin Nn
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wrex 2615  copk 4057   Nn cnnc 4373   cplc 4375   fin clefin 4432 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-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-rex 2620  df-v 2861  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-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-addc 4378  df-lefin 4440 This theorem is referenced by:  lefinaddc  4450  nulge  4456  leltfintr  4458  lefinlteq  4463  ltfintri  4466  lefinrflx  4467  ltlefin  4468  vfinspsslem1  4550
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