Theorem altopth2 36109

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

Assertion

Ref	Expression
altopth2	⊢ (𝐵 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))

Proof of Theorem altopth2

Step	Hyp	Ref	Expression
1		altopthsn 36104	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqrg 4793	. . 3 ⊢ (𝐵 ∈ 𝑉 → ({𝐵} = {𝐷} → 𝐵 = 𝐷))
3	2	adantld 490	. 2 ⊢ (𝐵 ∈ 𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐵 = 𝐷))
4	1, 3	biimtrid 242	1 ⊢ (𝐵 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {csn 4578  ⟪caltop 36099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-altop 36101

This theorem is referenced by: (None)