Theorem altopthb 36014

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 36013 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthb.1	⊢ 𝐴 ∈ V
altopthb.2	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
altopthb	⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem altopthb

Step	Hyp	Ref	Expression
1		altopthb.1	. 2 ⊢ 𝐴 ∈ V
2		altopthb.2	. 2 ⊢ 𝐷 ∈ V
3		altopthbg 36012	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 692	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟪caltop 36000

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-altop 36002

This theorem is referenced by:  altopthc  36015