Theorem axtyplowerprim 4095

Description: ax-typlower 4087 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axtyplowerprim	⊢ ∃y∀z(z ∈ y ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x))

Distinct variable groups:   a,b,w   x,a,z   b,c   b,d,w,z   w,c   e,d   w,d,z   z,e   x,w,y,z

Proof of Theorem axtyplowerprim

Step	Hyp	Ref	Expression
1		ax-typlower 4087	. 2 ⊢ ∃y∀z(z ∈ y ↔ ∀w⟪w, {z}⟫ ∈ x)
2		df-clel 2349	. . . . . . 7 ⊢ (⟪w, {z}⟫ ∈ x ↔ ∃a(a = ⟪w, {z}⟫ ∧ a ∈ x))
3		axprimlem2 4090	. . . . . . . . . 10 ⊢ (a = ⟪w, {z}⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ d = {z})))))
4		axprimlem1 4089	. . . . . . . . . . . . . . . 16 ⊢ (d = {z} ↔ ∀e(e ∈ d ↔ e = z))
5	4	orbi2i 505	. . . . . . . . . . . . . . 15 ⊢ ((d = w ∨ d = {z}) ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z)))
6	5	bibi2i 304	. . . . . . . . . . . . . 14 ⊢ ((d ∈ b ↔ (d = w ∨ d = {z})) ↔ (d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))
7	6	albii 1566	. . . . . . . . . . . . 13 ⊢ (∀d(d ∈ b ↔ (d = w ∨ d = {z})) ↔ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))
8	7	orbi2i 505	. . . . . . . . . . . 12 ⊢ ((∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ d = {z}))) ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z)))))
9	8	bibi2i 304	. . . . . . . . . . 11 ⊢ ((b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ d = {z})))) ↔ (b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))))
10	9	albii 1566	. . . . . . . . . 10 ⊢ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ d = {z})))) ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))))
11	3, 10	bitri 240	. . . . . . . . 9 ⊢ (a = ⟪w, {z}⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))))
12	11	anbi1i 676	. . . . . . . 8 ⊢ ((a = ⟪w, {z}⟫ ∧ a ∈ x) ↔ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x))
13	12	exbii 1582	. . . . . . 7 ⊢ (∃a(a = ⟪w, {z}⟫ ∧ a ∈ x) ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x))
14	2, 13	bitri 240	. . . . . 6 ⊢ (⟪w, {z}⟫ ∈ x ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x))
15	14	albii 1566	. . . . 5 ⊢ (∀w⟪w, {z}⟫ ∈ x ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x))
16	15	bibi2i 304	. . . 4 ⊢ ((z ∈ y ↔ ∀w⟪w, {z}⟫ ∈ x) ↔ (z ∈ y ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x)))
17	16	albii 1566	. . 3 ⊢ (∀z(z ∈ y ↔ ∀w⟪w, {z}⟫ ∈ x) ↔ ∀z(z ∈ y ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x)))
18	17	exbii 1582	. 2 ⊢ (∃y∀z(z ∈ y ↔ ∀w⟪w, {z}⟫ ∈ x) ↔ ∃y∀z(z ∈ y ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ x)))
19	1, 18	mpbi 199	1 ⊢ ∃y∀z(z ∈ y ↔ ∀w∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = w) ∨ ∀d(d ∈ b ↔ (d = w ∨ ∀e(e ∈ d ↔ e = z))))) ∧ a ∈ 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-typlower 4087

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)