Theorem altopthg 36732

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopthg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem altopthg

Step	Hyp	Ref	Expression
1		altopthsn 36726	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4802	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4802	. . 3 ⊢ (𝐵 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
4	2, 3	bi2anan9 650	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
5	1, 4	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {csn 4583  ⟪caltop 36721

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-sn 4584  df-pr 4586  df-altop 36723

This theorem is used by:  altopth  36734