Theorem altopth2 33851

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 33846	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqrg 4730	. . 3 ⊢ (𝐵 ∈ 𝑉 → ({𝐵} = {𝐷} → 𝐵 = 𝐷))
3	2	adantld 494	. 2 ⊢ (𝐵 ∈ 𝑉 → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) → 𝐵 = 𝐷))
4	1, 3	syl5bi 245	1 ⊢ (𝐵 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525  ⟪caltop 33841

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-altop 33843

This theorem is referenced by: (None)