Theorem opklefing 4449

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 4441	. 2 ⊢ ≤fin = {w ∣ ∃y∃z(w = ⟪y, z⟫ ∧ ∃x ∈ Nn z = (y +c x))}
2		addceq1 4384	. . . 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 2636	. 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 2636	. 2 ⊢ (z = B → (∃x ∈ Nn z = (A +c x) ↔ ∃x ∈ Nn B = (A +c x)))
7	1, 4, 6	opkelopkabg 4246	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 2616  ⟪copk 4058   Nn cnnc 4374   +_c cplc 4376   ≤_fin clefin 4433

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-addc 4379  df-lefin 4441

This theorem is referenced by:  lefinaddc  4451  nulge  4457  leltfintr  4459  lefinlteq  4464  ltfintri  4467  lefinrflx  4468  ltlefin  4469  vfinspsslem1  4551