Theorem axins3prim 4097

Description: ax-ins3 4086 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axins3prim	|- E.yA.zA.wA.t(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x))

Distinct variable groups:   a,b,t,w   y,a,z   b,c   f,b,t,w,z   c,d,z   e,d,z   e,g,z   f,g   f,h,t,w   z,f,g   h,i   h,j,t,w   w,i   t,j,w   k,l,w   x,k,z   m,l   n,l,w,z   z,m   w,n,z   w,t,x,y,z

Proof of Theorem axins3prim

Step	Hyp	Ref	Expression
1		ax-ins3 4086	. 2 |- E.yA.zA.wA.t(<<{{z}}, <<w, t>>>> e. y <-> <<z, w>> e. x)
2		df-clel 2349	. . . . . . 7 |- (<<{{z}}, <<w, t>>>> e. y <-> E.a(a = <<{{z}}, <<w, t>>>> /\ a e. y))
3		axprimlem2 4090	. . . . . . . . . 10 |- (a = <<{{z}}, <<w, t>>>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = {{z}}) \/ A.f(f e. b <-> (f = {{z}} \/ f = <<w, t>>)))))
4		axprimlem1 4089	. . . . . . . . . . . . . . . 16 |- (c = {{z}} <-> A.d(d e. c <-> d = {z}))
5		axprimlem1 4089	. . . . . . . . . . . . . . . . . 18 |- (d = {z} <-> A.e(e e. d <-> e = z))
6	5	bibi2i 304	. . . . . . . . . . . . . . . . 17 |- ((d e. c <-> d = {z}) <-> (d e. c <-> A.e(e e. d <-> e = z)))
7	6	albii 1566	. . . . . . . . . . . . . . . 16 |- (A.d(d e. c <-> d = {z}) <-> A.d(d e. c <-> A.e(e e. d <-> e = z)))
8	4, 7	bitri 240	. . . . . . . . . . . . . . 15 |- (c = {{z}} <-> A.d(d e. c <-> A.e(e e. d <-> e = z)))
9	8	bibi2i 304	. . . . . . . . . . . . . 14 |- ((c e. b <-> c = {{z}}) <-> (c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))))
10	9	albii 1566	. . . . . . . . . . . . 13 |- (A.c(c e. b <-> c = {{z}}) <-> A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))))
11		axprimlem1 4089	. . . . . . . . . . . . . . . . 17 |- (f = {{z}} <-> A.g(g e. f <-> g = {z}))
12		axprimlem1 4089	. . . . . . . . . . . . . . . . . . 19 |- (g = {z} <-> A.e(e e. g <-> e = z))
13	12	bibi2i 304	. . . . . . . . . . . . . . . . . 18 |- ((g e. f <-> g = {z}) <-> (g e. f <-> A.e(e e. g <-> e = z)))
14	13	albii 1566	. . . . . . . . . . . . . . . . 17 |- (A.g(g e. f <-> g = {z}) <-> A.g(g e. f <-> A.e(e e. g <-> e = z)))
15	11, 14	bitri 240	. . . . . . . . . . . . . . . 16 |- (f = {{z}} <-> A.g(g e. f <-> A.e(e e. g <-> e = z)))
16		axprimlem2 4090	. . . . . . . . . . . . . . . 16 |- (f = <<w, t>> <-> A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))
17	15, 16	orbi12i 507	. . . . . . . . . . . . . . 15 |- ((f = {{z}} \/ f = <<w, t>>) <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t))))))
18	17	bibi2i 304	. . . . . . . . . . . . . 14 |- ((f e. b <-> (f = {{z}} \/ f = <<w, t>>)) <-> (f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))
19	18	albii 1566	. . . . . . . . . . . . 13 |- (A.f(f e. b <-> (f = {{z}} \/ f = <<w, t>>)) <-> A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))
20	10, 19	orbi12i 507	. . . . . . . . . . . 12 |- ((A.c(c e. b <-> c = {{z}}) \/ A.f(f e. b <-> (f = {{z}} \/ f = <<w, t>>))) <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t))))))))
21	20	bibi2i 304	. . . . . . . . . . 11 |- ((b e. a <-> (A.c(c e. b <-> c = {{z}}) \/ A.f(f e. b <-> (f = {{z}} \/ f = <<w, t>>)))) <-> (b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))))
22	21	albii 1566	. . . . . . . . . 10 |- (A.b(b e. a <-> (A.c(c e. b <-> c = {{z}}) \/ A.f(f e. b <-> (f = {{z}} \/ f = <<w, t>>)))) <-> A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))))
23	3, 22	bitri 240	. . . . . . . . 9 |- (a = <<{{z}}, <<w, t>>>> <-> A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))))
24	23	anbi1i 676	. . . . . . . 8 |- ((a = <<{{z}}, <<w, t>>>> /\ a e. y) <-> (A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y))
25	24	exbii 1582	. . . . . . 7 |- (E.a(a = <<{{z}}, <<w, t>>>> /\ a e. y) <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y))
26	2, 25	bitri 240	. . . . . 6 |- (<<{{z}}, <<w, t>>>> e. y <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y))
27		df-clel 2349	. . . . . . 7 |- (<<z, w>> e. x <-> E.k(k = <<z, w>> /\ k e. x))
28		axprimlem2 4090	. . . . . . . . 9 |- (k = <<z, w>> <-> A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))))
29	28	anbi1i 676	. . . . . . . 8 |- ((k = <<z, w>> /\ k e. x) <-> (A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x))
30	29	exbii 1582	. . . . . . 7 |- (E.k(k = <<z, w>> /\ k e. x) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x))
31	27, 30	bitri 240	. . . . . 6 |- (<<z, w>> e. x <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x))
32	26, 31	bibi12i 306	. . . . 5 |- ((<<{{z}}, <<w, t>>>> e. y <-> <<z, w>> e. x) <-> (E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x)))
33	32	albii 1566	. . . 4 |- (A.t(<<{{z}}, <<w, t>>>> e. y <-> <<z, w>> e. x) <-> A.t(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x)))
34	33	2albii 1567	. . 3 |- (A.zA.wA.t(<<{{z}}, <<w, t>>>> e. y <-> <<z, w>> e. x) <-> A.zA.wA.t(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x)))
35	34	exbii 1582	. 2 |- (E.yA.zA.wA.t(<<{{z}}, <<w, t>>>> e. y <-> <<z, w>> e. x) <-> E.yA.zA.wA.t(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x)))
36	1, 35	mpbi 199	1 |- E.yA.zA.wA.t(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> A.e(e e. d <-> e = z))) \/ A.f(f e. b <-> (A.g(g e. f <-> A.e(e e. g <-> e = z)) \/ A.h(h e. f <-> (A.i(i e. h <-> i = w) \/ A.j(j e. h <-> (j = w \/ j = t)))))))) /\ a e. y) <-> E.k(A.l(l e. k <-> (A.m(m e. l <-> m = z) \/ A.n(n e. l <-> (n = z \/ n = w)))) /\ k e. x))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {csn 3738  <<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  ax-ins3 4086

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: (None)