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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 ((A V B W) → (⟪A, Bfinx Nn B = (A +c x)))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   V(x)   W(x)

Proof of Theorem opklefing
Dummy variables y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lefin 4440 . 2 fin = {w yz(w = ⟪y, z x Nn z = (y +c x))}
2 addceq1 4383 . . . 4 (y = A → (y +c x) = (A +c x))
32eqeq2d 2364 . . 3 (y = A → (z = (y +c x) ↔ z = (A +c x)))
43rexbidv 2635 . 2 (y = A → (x Nn z = (y +c x) ↔ x Nn z = (A +c x)))
5 eqeq1 2359 . . 3 (z = B → (z = (A +c x) ↔ B = (A +c x)))
65rexbidv 2635 . 2 (z = B → (x Nn z = (A +c x) ↔ x Nn B = (A +c x)))
71, 4, 6opkelopkabg 4245 1 ((A V B W) → (⟪A, Bfinx Nn B = (A +c x)))
 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   +c cplc 4375   ≤fin clefin 4432 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-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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