Theorem altopth1 36530

Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopth1	⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))

Proof of Theorem altopth1

Step	Hyp	Ref	Expression
1		altopthsn 36526	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqrg 4802	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
3	2	adantrd 497	. 2 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐴 = 𝐶))
4	1, 3	biimtrid 245	1 ⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4587  ⟪caltop 36521

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-altop 36523

This theorem is used by: (None)