Theorem axprimlem2 4090

Description: Lemma for the primitive axioms. Primitive form of equality to a Kuratowski ordered pair. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axprimlem2	|- (a = <<B, C>> <-> A.d(d e. a <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C)))))

Distinct variable groups:   a,d   B,d,e   B,f   C,d,f   e,d   f,d

Allowed substitution hints:   B(a)   C(e,a)

Proof of Theorem axprimlem2

Step	Hyp	Ref	Expression
1		df-opk 4059	. . 3 |- <<B, C>> = {{B}, {B, C}}
2	1	eqeq2i 2363	. 2 |- (a = <<B, C>> <-> a = {{B}, {B, C}})
3		dfcleq 2347	. . 3 |- (a = {{B}, {B, C}} <-> A.d(d e. a <-> d e. {{B}, {B, C}}))
4		vex 2863	. . . . . . 7 |- d e. _V
5	4	elpr 3752	. . . . . 6 |- (d e. {{B}, {B, C}} <-> (d = {B} \/ d = {B, C}))
6		axprimlem1 4089	. . . . . . 7 |- (d = {B} <-> A.e(e e. d <-> e = B))
7		dfcleq 2347	. . . . . . . 8 |- (d = {B, C} <-> A.f(f e. d <-> f e. {B, C}))
8		vex 2863	. . . . . . . . . . 11 |- f e. _V
9	8	elpr 3752	. . . . . . . . . 10 |- (f e. {B, C} <-> (f = B \/ f = C))
10	9	bibi2i 304	. . . . . . . . 9 |- ((f e. d <-> f e. {B, C}) <-> (f e. d <-> (f = B \/ f = C)))
11	10	albii 1566	. . . . . . . 8 |- (A.f(f e. d <-> f e. {B, C}) <-> A.f(f e. d <-> (f = B \/ f = C)))
12	7, 11	bitri 240	. . . . . . 7 |- (d = {B, C} <-> A.f(f e. d <-> (f = B \/ f = C)))
13	6, 12	orbi12i 507	. . . . . 6 |- ((d = {B} \/ d = {B, C}) <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C))))
14	5, 13	bitri 240	. . . . 5 |- (d e. {{B}, {B, C}} <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C))))
15	14	bibi2i 304	. . . 4 |- ((d e. a <-> d e. {{B}, {B, C}}) <-> (d e. a <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C)))))
16	15	albii 1566	. . 3 |- (A.d(d e. a <-> d e. {{B}, {B, C}}) <-> A.d(d e. a <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C)))))
17	3, 16	bitri 240	. 2 |- (a = {{B}, {B, C}} <-> A.d(d e. a <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C)))))
18	2, 17	bitri 240	1 |- (a = <<B, C>> <-> A.d(d e. a <-> (A.e(e e. d <-> e = B) \/ A.f(f e. d <-> (f = B \/ f = C)))))

Syntax hints:   <-> wb 176   \/ wo 357  A.wal 1540   = wceq 1642   e. wcel 1710  {csn 3738  {cpr 3739  <<copk 4058

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  axxpprim  4091  axcnvprim  4092  axssetprim  4093  axsiprim  4094  axtyplowerprim  4095  axins2prim  4096  axins3prim  4097