Theorem axsiprim 4094

Description: ax-si 4084 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

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

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

Proof of Theorem axsiprim

Step	Hyp	Ref	Expression
1		ax-si 4084	. 2 |- E.yA.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x)
2		df-clel 2349	. . . . . 6 |- (<<{z}, {w}>> e. y <-> E.a(a = <<{z}, {w}>> /\ a e. y))
3		axprimlem2 4090	. . . . . . . . 9 |- (a = <<{z}, {w}>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = {z}) \/ A.e(e e. b <-> (e = {z} \/ e = {w})))))
4		axprimlem1 4089	. . . . . . . . . . . . . 14 |- (c = {z} <-> A.d(d e. c <-> d = z))
5	4	bibi2i 304	. . . . . . . . . . . . 13 |- ((c e. b <-> c = {z}) <-> (c e. b <-> A.d(d e. c <-> d = z)))
6	5	albii 1566	. . . . . . . . . . . 12 |- (A.c(c e. b <-> c = {z}) <-> A.c(c e. b <-> A.d(d e. c <-> d = z)))
7		axprimlem1 4089	. . . . . . . . . . . . . . 15 |- (e = {z} <-> A.f(f e. e <-> f = z))
8		axprimlem1 4089	. . . . . . . . . . . . . . 15 |- (e = {w} <-> A.g(g e. e <-> g = w))
9	7, 8	orbi12i 507	. . . . . . . . . . . . . 14 |- ((e = {z} \/ e = {w}) <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w)))
10	9	bibi2i 304	. . . . . . . . . . . . 13 |- ((e e. b <-> (e = {z} \/ e = {w})) <-> (e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))
11	10	albii 1566	. . . . . . . . . . . 12 |- (A.e(e e. b <-> (e = {z} \/ e = {w})) <-> A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))
12	6, 11	orbi12i 507	. . . . . . . . . . 11 |- ((A.c(c e. b <-> c = {z}) \/ A.e(e e. b <-> (e = {z} \/ e = {w}))) <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w)))))
13	12	bibi2i 304	. . . . . . . . . 10 |- ((b e. a <-> (A.c(c e. b <-> c = {z}) \/ A.e(e e. b <-> (e = {z} \/ e = {w})))) <-> (b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))))
14	13	albii 1566	. . . . . . . . 9 |- (A.b(b e. a <-> (A.c(c e. b <-> c = {z}) \/ A.e(e e. b <-> (e = {z} \/ e = {w})))) <-> A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))))
15	3, 14	bitri 240	. . . . . . . 8 |- (a = <<{z}, {w}>> <-> A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))))
16	15	anbi1i 676	. . . . . . 7 |- ((a = <<{z}, {w}>> /\ a e. y) <-> (A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y))
17	16	exbii 1582	. . . . . 6 |- (E.a(a = <<{z}, {w}>> /\ a e. y) <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y))
18	2, 17	bitri 240	. . . . 5 |- (<<{z}, {w}>> e. y <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y))
19		df-clel 2349	. . . . . 6 |- (<<z, w>> e. x <-> E.h(h = <<z, w>> /\ h e. x))
20		axprimlem2 4090	. . . . . . . 8 |- (h = <<z, w>> <-> A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))))
21	20	anbi1i 676	. . . . . . 7 |- ((h = <<z, w>> /\ h e. x) <-> (A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x))
22	21	exbii 1582	. . . . . 6 |- (E.h(h = <<z, w>> /\ h e. x) <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x))
23	19, 22	bitri 240	. . . . 5 |- (<<z, w>> e. x <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x))
24	18, 23	bibi12i 306	. . . 4 |- ((<<{z}, {w}>> e. y <-> <<z, w>> e. x) <-> (E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y) <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x)))
25	24	2albii 1567	. . 3 |- (A.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x) <-> A.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y) <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x)))
26	25	exbii 1582	. 2 |- (E.yA.zA.w(<<{z}, {w}>> e. y <-> <<z, w>> e. x) <-> E.yA.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y) <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h e. x)))
27	1, 26	mpbi 199	1 |- E.yA.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> A.d(d e. c <-> d = z)) \/ A.e(e e. b <-> (A.f(f e. e <-> f = z) \/ A.g(g e. e <-> g = w))))) /\ a e. y) <-> E.h(A.i(i e. h <-> (A.j(j e. i <-> j = z) \/ A.k(k e. i <-> (k = z \/ k = w)))) /\ h 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-si 4084

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)