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Theorem opkltfing 4449
 Description: Kuratowski ordered pair membership in finite less than. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
opkltfing fin Nn 1c
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem opkltfing
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltfin 4441 . 2 fin Nn 1c
2 neeq1 2524 . . 3
3 addceq1 4383 . . . . . 6
43addceq1d 4389 . . . . 5 1c 1c
54eqeq2d 2364 . . . 4 1c 1c
65rexbidv 2635 . . 3 Nn 1c Nn 1c
72, 6anbi12d 691 . 2 Nn 1c Nn 1c
8 eqeq1 2359 . . . 4 1c 1c
98rexbidv 2635 . . 3 Nn 1c Nn 1c
109anbi2d 684 . 2 Nn 1c Nn 1c
111, 7, 10opkelopkabg 4245 1 fin Nn 1c
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710   wne 2516  wrex 2615  c0 3550  copk 4057  1cc1c 4134   Nn cnnc 4373   cplc 4375   fin cltfin 4433 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ltfin 4441 This theorem is referenced by:  ltfinirr  4457  leltfintr  4458  ltfintr  4459  ltfinp1  4462  lefinlteq  4463  ltfintri  4466  ltlefin  4468  tfinltfinlem1  4500  tfinltfin  4501  sfinltfin  4535
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