Theorem oteqex 2806

Description: Equivalence of existence implied by equality of ordered triples.

Assertion

Ref	Expression
oteqex	⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨R, S⟩, T⟩ → (A ∈ V ↔ R ∈ V))

Proof of Theorem oteqex

Step	Hyp	Ref	Expression
1		opex 2789	. . 3 ⊢ ⟨A, B⟩ ∈ V
2	1	opth1 2793	. 2 ⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨R, S⟩, T⟩ → ⟨A, B⟩ = ⟨R, S⟩)
3		opeqex 2805	. 2 ⊢ (⟨A, B⟩ = ⟨R, S⟩ → (A ∈ V ↔ R ∈ V))
4	2, 3	syl 10	1 ⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨R, S⟩, T⟩ → (A ∈ V ↔ R ∈ V))

Syntax hints:   → wi 3   ↔ wb 146   = wceq 958   ∈ wcel 960  Vcvv 1814  ⟨cop 2416

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421

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