Theorem axssetprim 4093

Description: ax-sset 4083 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

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

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

Proof of Theorem axssetprim

Step	Hyp	Ref	Expression
1		ax-sset 4083	. 2 ⊢ ∃x∀y∀z(⟪y, z⟫ ∈ x ↔ ∀e(e ∈ y → e ∈ z))
2		df-clel 2349	. . . . . 6 ⊢ (⟪y, z⟫ ∈ x ↔ ∃a(a = ⟪y, z⟫ ∧ a ∈ x))
3		axprimlem2 4090	. . . . . . . 8 ⊢ (a = ⟪y, z⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))))
4	3	anbi1i 676	. . . . . . 7 ⊢ ((a = ⟪y, z⟫ ∧ a ∈ x) ↔ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x))
5	4	exbii 1582	. . . . . 6 ⊢ (∃a(a = ⟪y, z⟫ ∧ a ∈ x) ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x))
6	2, 5	bitri 240	. . . . 5 ⊢ (⟪y, z⟫ ∈ x ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x))
7	6	bibi1i 305	. . . 4 ⊢ ((⟪y, z⟫ ∈ x ↔ ∀e(e ∈ y → e ∈ z)) ↔ (∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x) ↔ ∀e(e ∈ y → e ∈ z)))
8	7	2albii 1567	. . 3 ⊢ (∀y∀z(⟪y, z⟫ ∈ x ↔ ∀e(e ∈ y → e ∈ z)) ↔ ∀y∀z(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x) ↔ ∀e(e ∈ y → e ∈ z)))
9	8	exbii 1582	. 2 ⊢ (∃x∀y∀z(⟪y, z⟫ ∈ x ↔ ∀e(e ∈ y → e ∈ z)) ↔ ∃x∀y∀z(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x) ↔ ∀e(e ∈ y → e ∈ z)))
10	1, 9	mpbi 199	1 ⊢ ∃x∀y∀z(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = y) ∨ ∀d(d ∈ b ↔ (d = y ∨ d = z)))) ∧ a ∈ x) ↔ ∀e(e ∈ y → e ∈ z))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟪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-sset 4083

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)