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, B⟫ ∈ ≤fin ↔ ∃x ∈ 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.

Step	Hyp	Ref	Expression
1		df-lefin 4440	. 2 ⊢ ≤fin = {w ∣ ∃y∃z(w = ⟪y, z⟫ ∧ ∃x ∈ Nn z = (y +c x))}
2		addceq1 4383	. . . 4 ⊢ (y = A → (y +c x) = (A +c x))
3	2	eqeq2d 2364	. . 3 ⊢ (y = A → (z = (y +c x) ↔ z = (A +c x)))
4	3	rexbidv 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)))
6	5	rexbidv 2635	. 2 ⊢ (z = B → (∃x ∈ Nn z = (A +c x) ↔ ∃x ∈ Nn B = (A +c x)))
7	1, 4, 6	opkelopkabg 4245	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ ≤fin ↔ ∃x ∈ Nn B = (A +c x)))

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-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