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Theorem axprimlem2 4089
 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 de = 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
StepHypRef Expression
1 df-opk 4058 . . 3 B, C⟫ = {{B}, {B, C}}
21eqeq2i 2363 . 2 (a = ⟪B, C⟫ ↔ a = {{B}, {B, C}})
3 dfcleq 2347 . . 3 (a = {{B}, {B, C}} ↔ d(d ad {{B}, {B, C}}))
4 vex 2862 . . . . . . 7 d V
54elpr 3751 . . . . . 6 (d {{B}, {B, C}} ↔ (d = {B} d = {B, C}))
6 axprimlem1 4088 . . . . . . 7 (d = {B} ↔ e(e de = B))
7 dfcleq 2347 . . . . . . . 8 (d = {B, C} ↔ f(f df {B, C}))
8 vex 2862 . . . . . . . . . . 11 f V
98elpr 3751 . . . . . . . . . 10 (f {B, C} ↔ (f = B f = C))
109bibi2i 304 . . . . . . . . 9 ((f df {B, C}) ↔ (f d ↔ (f = B f = C)))
1110albii 1566 . . . . . . . 8 (f(f df {B, C}) ↔ f(f d ↔ (f = B f = C)))
127, 11bitri 240 . . . . . . 7 (d = {B, C} ↔ f(f d ↔ (f = B f = C)))
136, 12orbi12i 507 . . . . . 6 ((d = {B} d = {B, C}) ↔ (e(e de = B) f(f d ↔ (f = B f = C))))
145, 13bitri 240 . . . . 5 (d {{B}, {B, C}} ↔ (e(e de = B) f(f d ↔ (f = B f = C))))
1514bibi2i 304 . . . 4 ((d ad {{B}, {B, C}}) ↔ (d a ↔ (e(e de = B) f(f d ↔ (f = B f = C)))))
1615albii 1566 . . 3 (d(d ad {{B}, {B, C}}) ↔ d(d a ↔ (e(e de = B) f(f d ↔ (f = B f = C)))))
173, 16bitri 240 . 2 (a = {{B}, {B, C}} ↔ d(d a ↔ (e(e de = B) f(f d ↔ (f = B f = C)))))
182, 17bitri 240 1 (a = ⟪B, C⟫ ↔ d(d a ↔ (e(e de = B) f(f d ↔ (f = B f = C)))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {csn 3737  {cpr 3738  ⟪copk 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by:  axxpprim  4090  axcnvprim  4091  axssetprim  4092  axsiprim  4093  axtyplowerprim  4094  axins2prim  4095  axins3prim  4096
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