Theorem ssexOLD 5297

Description: Obsolete version of ssex 5296 as of 18-Jul-2026. (Contributed by NM, 27-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ssex.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ssexOLD	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Proof of Theorem ssexOLD

Step	Hyp	Ref	Expression
1		dfss2 3926	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		ssex.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	inex2 5292	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
4		eleq1 2854	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 236	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V)
6	1, 5	sylbi 220	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925

This theorem is used by: (None)