Theorem opth2 4380

Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Hypotheses

Ref	Expression
opth2.1	⊢ 𝐶 ∈ V
opth2.2	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
opth2	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opth2.1	. 2 ⊢ 𝐶 ∈ V
2		opth2.2	. 2 ⊢ 𝐷 ∈ V
3		opthg2 4379	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 430	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718

This theorem is used by:  eqvinop  4383  opelxp  4804  fsn  5880  dfplpq2  7721  ltresr  8206  frecuzrdgtcl  10849  frecuzrdgfunlem  10856