Theorem axprimlem2 4090

Description: Lemma for the primitive axioms. Primitive form of equality to a Kuratowski ordered pair. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axprimlem2	⊢ (a = ⟪B, C⟫ ↔ ∀d(d ∈ a ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C)))))

Distinct variable groups:   a,d   B,d,e   B,f   C,d,f   e,d   f,d

Allowed substitution hints:   B(a)   C(e,a)

Proof of Theorem axprimlem2

Step	Hyp	Ref	Expression
1		df-opk 4059	. . 3 ⊢ ⟪B, C⟫ = {{B}, {B, C}}
2	1	eqeq2i 2363	. 2 ⊢ (a = ⟪B, C⟫ ↔ a = {{B}, {B, C}})
3		dfcleq 2347	. . 3 ⊢ (a = {{B}, {B, C}} ↔ ∀d(d ∈ a ↔ d ∈ {{B}, {B, C}}))
4		vex 2863	. . . . . . 7 ⊢ d ∈ V
5	4	elpr 3752	. . . . . 6 ⊢ (d ∈ {{B}, {B, C}} ↔ (d = {B} ∨ d = {B, C}))
6		axprimlem1 4089	. . . . . . 7 ⊢ (d = {B} ↔ ∀e(e ∈ d ↔ e = B))
7		dfcleq 2347	. . . . . . . 8 ⊢ (d = {B, C} ↔ ∀f(f ∈ d ↔ f ∈ {B, C}))
8		vex 2863	. . . . . . . . . . 11 ⊢ f ∈ V
9	8	elpr 3752	. . . . . . . . . 10 ⊢ (f ∈ {B, C} ↔ (f = B ∨ f = C))
10	9	bibi2i 304	. . . . . . . . 9 ⊢ ((f ∈ d ↔ f ∈ {B, C}) ↔ (f ∈ d ↔ (f = B ∨ f = C)))
11	10	albii 1566	. . . . . . . 8 ⊢ (∀f(f ∈ d ↔ f ∈ {B, C}) ↔ ∀f(f ∈ d ↔ (f = B ∨ f = C)))
12	7, 11	bitri 240	. . . . . . 7 ⊢ (d = {B, C} ↔ ∀f(f ∈ d ↔ (f = B ∨ f = C)))
13	6, 12	orbi12i 507	. . . . . 6 ⊢ ((d = {B} ∨ d = {B, C}) ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C))))
14	5, 13	bitri 240	. . . . 5 ⊢ (d ∈ {{B}, {B, C}} ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C))))
15	14	bibi2i 304	. . . 4 ⊢ ((d ∈ a ↔ d ∈ {{B}, {B, C}}) ↔ (d ∈ a ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C)))))
16	15	albii 1566	. . 3 ⊢ (∀d(d ∈ a ↔ d ∈ {{B}, {B, C}}) ↔ ∀d(d ∈ a ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C)))))
17	3, 16	bitri 240	. 2 ⊢ (a = {{B}, {B, C}} ↔ ∀d(d ∈ a ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C)))))
18	2, 17	bitri 240	1 ⊢ (a = ⟪B, C⟫ ↔ ∀d(d ∈ a ↔ (∀e(e ∈ d ↔ e = B) ∨ ∀f(f ∈ d ↔ (f = B ∨ f = C)))))

Syntax hints:   ↔ wb 176   ∨ wo 357  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {csn 3738  {cpr 3739  ⟪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

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:  axxpprim  4091  axcnvprim  4092  axssetprim  4093  axsiprim  4094  axtyplowerprim  4095  axins2prim  4096  axins3prim  4097