Theorem opkltfing 4450

Description: Kuratowski ordered pair membership in finite less than. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
opkltfing	|- ((A e. V /\ B e. W) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c))))

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   V(x)   W(x)

Proof of Theorem opkltfing

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltfin 4442	. 2 |- <fin = {y | E.zE.w(y = <<z, w>> /\ (z =/= (/) /\ E.x e. Nn w = ((z +o x) +o 1c)))}
2		neeq1 2525	. . 3 |- (z = A -> (z =/= (/) <-> A =/= (/)))
3		addceq1 4384	. . . . . 6 |- (z = A -> (z +o x) = (A +o x))
4	3	addceq1d 4390	. . . . 5 |- (z = A -> ((z +o x) +o 1c) = ((A +o x) +o 1c))
5	4	eqeq2d 2364	. . . 4 |- (z = A -> (w = ((z +o x) +o 1c) <-> w = ((A +o x) +o 1c)))
6	5	rexbidv 2636	. . 3 |- (z = A -> (E.x e. Nn w = ((z +o x) +o 1c) <-> E.x e. Nn w = ((A +o x) +o 1c)))
7	2, 6	anbi12d 691	. 2 |- (z = A -> ((z =/= (/) /\ E.x e. Nn w = ((z +o x) +o 1c)) <-> (A =/= (/) /\ E.x e. Nn w = ((A +o x) +o 1c))))
8		eqeq1 2359	. . . 4 |- (w = B -> (w = ((A +o x) +o 1c) <-> B = ((A +o x) +o 1c)))
9	8	rexbidv 2636	. . 3 |- (w = B -> (E.x e. Nn w = ((A +o x) +o 1c) <-> E.x e. Nn B = ((A +o x) +o 1c)))
10	9	anbi2d 684	. 2 |- (w = B -> ((A =/= (/) /\ E.x e. Nn w = ((A +o x) +o 1c)) <-> (A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c))))
11	1, 7, 10	opkelopkabg 4246	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c))))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  (/)c0 3551  <<copk 4058  1_cc1c 4135   Nn cnnc 4374   +o cplc 4376   <_fin cltfin 4434

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-ltfin 4442

This theorem is referenced by:  ltfinirr  4458  leltfintr  4459  ltfintr  4460  ltfinp1  4463  lefinlteq  4464  ltfintri  4467  ltlefin  4469  tfinltfinlem1  4501  tfinltfin  4502  sfinltfin  4536