Theorem elexOLD 3483

Description: Obsolete version of elex 3482 as of 28-May-2025. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elexOLD	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)

Proof of Theorem elexOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpl 1864	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑥 𝑥 = 𝐴)
2		dfclel 2804	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
3		isset 3476	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4	1, 2, 3	3imtr4i 291	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464

This theorem is referenced by: (None)