Theorem eqop 4612

Description: Express equality to an ordered pair. (Contributed by SF, 6-Jan-2015.)

Assertion

Ref	Expression
eqop	⊢ (A = ⟨B, C⟩ ↔ ∀z(z ∈ A ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c}))))

Distinct variable groups:   z,A,t   z,B,t   t,C,z

Proof of Theorem eqop

Step	Hyp	Ref	Expression
1		dfcleq 2347	. 2 ⊢ (A = ⟨B, C⟩ ↔ ∀z(z ∈ A ↔ z ∈ ⟨B, C⟩))
2		df-op 4567	. . . . . . 7 ⊢ ⟨B, C⟩ = ({z ∣ ∃t ∈ B z = Phi t} ∪ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})})
3	2	eleq2i 2417	. . . . . 6 ⊢ (z ∈ ⟨B, C⟩ ↔ z ∈ ({z ∣ ∃t ∈ B z = Phi t} ∪ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})}))
4		elun 3221	. . . . . 6 ⊢ (z ∈ ({z ∣ ∃t ∈ B z = Phi t} ∪ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})}) ↔ (z ∈ {z ∣ ∃t ∈ B z = Phi t} ∨ z ∈ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})}))
5	3, 4	bitri 240	. . . . 5 ⊢ (z ∈ ⟨B, C⟩ ↔ (z ∈ {z ∣ ∃t ∈ B z = Phi t} ∨ z ∈ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})}))
6		abid 2341	. . . . . 6 ⊢ (z ∈ {z ∣ ∃t ∈ B z = Phi t} ↔ ∃t ∈ B z = Phi t)
7		abid 2341	. . . . . 6 ⊢ (z ∈ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})} ↔ ∃t ∈ C z = ( Phi t ∪ {0c}))
8	6, 7	orbi12i 507	. . . . 5 ⊢ ((z ∈ {z ∣ ∃t ∈ B z = Phi t} ∨ z ∈ {z ∣ ∃t ∈ C z = ( Phi t ∪ {0c})}) ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c})))
9	5, 8	bitri 240	. . . 4 ⊢ (z ∈ ⟨B, C⟩ ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c})))
10	9	bibi2i 304	. . 3 ⊢ ((z ∈ A ↔ z ∈ ⟨B, C⟩) ↔ (z ∈ A ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c}))))
11	10	albii 1566	. 2 ⊢ (∀z(z ∈ A ↔ z ∈ ⟨B, C⟩) ↔ ∀z(z ∈ A ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c}))))
12	1, 11	bitri 240	1 ⊢ (A = ⟨B, C⟩ ↔ ∀z(z ∈ A ↔ (∃t ∈ B z = Phi t ∨ ∃t ∈ C z = ( Phi t ∪ {0c}))))

Syntax hints:   ↔ wb 176   ∨ wo 357  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616   ∪ cun 3208  {csn 3738  0_cc0c 4375  ⟨cop 4562   Phi cphi 4563

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-op 4567

This theorem is referenced by:  setconslem3  4734  setconslem7  4738