Theorem altopthbg 36655

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Assertion

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

Proof of Theorem altopthbg

Step	Hyp	Ref	Expression
1		altopthsn 36648	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4802	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4802	. . . 4 ⊢ (𝐷 ∈ 𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4		eqcom 2767	. . . 4 ⊢ ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5		eqcom 2767	. . . 4 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
6	3, 4, 5	3bitr4g 317	. . 3 ⊢ (𝐷 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
7	2, 6	bi2anan9 650	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
8	1, 7	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4583  ⟪caltop 36643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  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 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3903  df-ss 3915  df-sn 4584  df-pr 4586  df-altop 36645

This theorem is used by:  altopthb  36657