Theorem axsiprim 4094

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

Assertion

Ref	Expression
axsiprim	⊢ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ 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 ⊢ ∃y∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x)
2		df-clel 2349	. . . . . 6 ⊢ (⟪{z}, {w}⟫ ∈ y ↔ ∃a(a = ⟪{z}, {w}⟫ ∧ a ∈ y))
3		axprimlem2 4090	. . . . . . . . 9 ⊢ (a = ⟪{z}, {w}⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = {z}) ∨ ∀e(e ∈ b ↔ (e = {z} ∨ e = {w})))))
4		axprimlem1 4089	. . . . . . . . . . . . . 14 ⊢ (c = {z} ↔ ∀d(d ∈ c ↔ d = z))
5	4	bibi2i 304	. . . . . . . . . . . . 13 ⊢ ((c ∈ b ↔ c = {z}) ↔ (c ∈ b ↔ ∀d(d ∈ c ↔ d = z)))
6	5	albii 1566	. . . . . . . . . . . 12 ⊢ (∀c(c ∈ b ↔ c = {z}) ↔ ∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)))
7		axprimlem1 4089	. . . . . . . . . . . . . . 15 ⊢ (e = {z} ↔ ∀f(f ∈ e ↔ f = z))
8		axprimlem1 4089	. . . . . . . . . . . . . . 15 ⊢ (e = {w} ↔ ∀g(g ∈ e ↔ g = w))
9	7, 8	orbi12i 507	. . . . . . . . . . . . . 14 ⊢ ((e = {z} ∨ e = {w}) ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w)))
10	9	bibi2i 304	. . . . . . . . . . . . 13 ⊢ ((e ∈ b ↔ (e = {z} ∨ e = {w})) ↔ (e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))
11	10	albii 1566	. . . . . . . . . . . 12 ⊢ (∀e(e ∈ b ↔ (e = {z} ∨ e = {w})) ↔ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))
12	6, 11	orbi12i 507	. . . . . . . . . . 11 ⊢ ((∀c(c ∈ b ↔ c = {z}) ∨ ∀e(e ∈ b ↔ (e = {z} ∨ e = {w}))) ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w)))))
13	12	bibi2i 304	. . . . . . . . . 10 ⊢ ((b ∈ a ↔ (∀c(c ∈ b ↔ c = {z}) ∨ ∀e(e ∈ b ↔ (e = {z} ∨ e = {w})))) ↔ (b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))))
14	13	albii 1566	. . . . . . . . 9 ⊢ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = {z}) ∨ ∀e(e ∈ b ↔ (e = {z} ∨ e = {w})))) ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))))
15	3, 14	bitri 240	. . . . . . . 8 ⊢ (a = ⟪{z}, {w}⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))))
16	15	anbi1i 676	. . . . . . 7 ⊢ ((a = ⟪{z}, {w}⟫ ∧ a ∈ y) ↔ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y))
17	16	exbii 1582	. . . . . 6 ⊢ (∃a(a = ⟪{z}, {w}⟫ ∧ a ∈ y) ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y))
18	2, 17	bitri 240	. . . . 5 ⊢ (⟪{z}, {w}⟫ ∈ y ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y))
19		df-clel 2349	. . . . . 6 ⊢ (⟪z, w⟫ ∈ x ↔ ∃h(h = ⟪z, w⟫ ∧ h ∈ x))
20		axprimlem2 4090	. . . . . . . 8 ⊢ (h = ⟪z, w⟫ ↔ ∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))))
21	20	anbi1i 676	. . . . . . 7 ⊢ ((h = ⟪z, w⟫ ∧ h ∈ x) ↔ (∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x))
22	21	exbii 1582	. . . . . 6 ⊢ (∃h(h = ⟪z, w⟫ ∧ h ∈ x) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x))
23	19, 22	bitri 240	. . . . 5 ⊢ (⟪z, w⟫ ∈ x ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x))
24	18, 23	bibi12i 306	. . . 4 ⊢ ((⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) ↔ (∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x)))
25	24	2albii 1567	. . 3 ⊢ (∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) ↔ ∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x)))
26	25	exbii 1582	. 2 ⊢ (∃y∀z∀w(⟪{z}, {w}⟫ ∈ y ↔ ⟪z, w⟫ ∈ x) ↔ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x)))
27	1, 26	mpbi 199	1 ⊢ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ ∀d(d ∈ c ↔ d = z)) ∨ ∀e(e ∈ b ↔ (∀f(f ∈ e ↔ f = z) ∨ ∀g(g ∈ e ↔ g = w))))) ∧ a ∈ y) ↔ ∃h(∀i(i ∈ h ↔ (∀j(j ∈ i ↔ j = z) ∨ ∀k(k ∈ i ↔ (k = z ∨ k = w)))) ∧ h ∈ x))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ 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)