Theorem altopth1 36187

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 36183	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqrg 4797	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
3	2	adantrd 491	. 2 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐴 = 𝐶))
4	1, 3	biimtrid 242	1 ⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4582  ⟪caltop 36178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-altop 36180

This theorem is referenced by: (None)